Coordination of Traffic Signals in Networks

Transcription

1 Coordination of Traffic Signals in Networks and Related Graph Theoretical Problems on Spanning Trees vorgelegt von Dipl.-Math. Gregor Wünsch Von der Fakultät II Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. eingereichte Dissertation Vorsitzender: Berichter: Prof. Dr. Rainer Wüst Prof. Dr. Rolf H. Möhring Prof. Dr. Ekkehard G. Köhler Berlin 2008 D 83

2

3 ACKNOWLEDGEMENTS This work would not have been possible without the help of a number of people. I wish to thank everybody who has been involved, directly or indirectly. First of all, I thank my supervisor Rolf Möhring. I am grateful for his support and encouragement and I am especially indebted to him raising my interest for discrete applied mathematics during my years as an undergraduate. In addition, I wish to thank Ekkehard Köhler for taking the second assessment of this thesis and for numerous discussions and stimulating ideas on traffic models involving traffic signals. Next, I wish to thank Klaus Nökel from PTV AG in Karlsruhe. During the last approximately four years he and the PTV supported the work on an optimization tool for coordinating signals in various ways. Not only that we were equipped with software, but also I enjoyed our discussions and profited from valuable suggestions. Also, I am very grateful for the financial support that I received within the DFG research training group MAGSI (GK-621). I thank all my colleagues from MAGSI for a fruitful interdisciplinary research environment. Moreover, I thank Armin Zimmermann for organizing almost everything concerning the activities within MAGSI. Furthermore, I am indebted to my coauthors Ekki Köhler, Rolf Möhring, Klaus Nökel, Alexander Reich and Romeo Rizzi and, especially, Christian Liebchen, who raised my interest for strictly fundamental cycle bases. Yet, from all of them, I learned a lot during intensive discussions. I also want to thank all members of the research groups of Rolf Möhring, Günter Ziegler and Stefan Felsner for the excellent working atmosphere. Here, special thanks go to my office mates Felix König, Heiko Schilling and Björn Stenzel. I very much enjoyed our numerous chats on various things like sports, elections, statistics and discrete mathematics. Moreover, I would like to thank Christian Liebchen, Nina Brenner, Tobias Harks, Richard Lütjens, Nicole Megow, Guido Schäfer, Izaskun Seara Tejados, Björn Stenzel, and Sebastian Stiller for their careful proofreading of the manuscript. The exposition of this thesis improved from their valuable suggestions and comments. Last, but surely not least, I am grateful to my friends and my family for their support and their interest in my work. Berlin, February 2008 Gregor Wünsch iii

4

5 CONTENTS Introduction 1 1 The Network Signal Coordination Problem Introduction to Coordination of Signals in Networks The Language of Traffic Signals Optimizing Traffic Lights Software to Optimize Coordination The Network Signal Coordination (NSC) Problem Definition of the NSC The Relation Between the NSC and the PESP NP-completeness of the c-nsc problem A Revised MIP Formulation for the NSC problem Introduction The Offsets The Cycle Equations A Variable Phase Sequencing The Objective Function Non-uniform Cycle-lengths The Mixed-Integer Linear Program Application of the NSC Model in Practice Conclusion and Open Questions Strictly Fundamental Cycle Bases on Grids Introduction Prelimenaries Lower Bounds A New Asymptotical Lower Bound The Challenge of Small Grids The Amaldi MIP for the General MSFCB Problem A new MIP formulation A Tight Bound for G 8, v

6 vi CONTENTS 2.4 Upper Bounds A New Asymptotical Upper Bound Experiments Conclusions and Open Questions Classification of Tree Spanner Problems Introduction A Unified Notation for Tree Spanners (UNTS) Maximum Stretch Problems Coincidences Anticoincidences Average Stretch Problems Coincidences Anticoincidences Max-Stretch And Average-Stretch Problems Never Coincide First Benefit of the UNTS An Open Complexity Status Inapproximability of the MMST Problem Conclusions and Open Questions Experiments A MIP Solver Comparison on Selected NSC Instances The Influence of Cycle Bases on the MIP Performance Case Studies Evaluating an NSC model Data Acquisition Portland Denver Conclusion and Open Questions Bibliography 143

7 INTRODUCTION Are you the type of person that likes waiting at a red light? Do you enjoy being stopped by uncoordinated signals? If not, you are invited to read on to find out how the coordination of traffic signals can help to reduce delays and, thus, avoid having to wait at red lights. In urban areas there is a strong demand for transportation. Probably, the most sustainable means of transportation are the public ones, like buses, trams or the underground. Nevertheless, not all demands for transportation can be covered by the public sector. A major part of the overall transportation in cities is composed of individual drivers. There are different ways to improve road traffic conditions in city areas. However, infrastructural arrangements like broadening streets or even building new ones to cope with increasing demands are often not an appropriate option, due to high costs or space limitations. Instead, intelligent means of traffic control are required to solve today s road traffic problems, like high delays, narrow capacities or traffic jams. When speaking of ways to control traffic, traffic lights or traffic signals are of primary importance. A clever adjustment of the signal settings surely helps to reduce delays and increase capacities, thereby avoiding traffic jams. Today, intelligent computer-aided traffic control signals are even capable of reacting to different traffic situations. Namely, they adapt their settings to the respective demands at the junctions. However, there are traffic scenarios where the traffic responsive signals reach their limit. For example, when there is constant and high traffic volume, the responsive signals repeatedly apply similar control strategies. Therefore, they behave comparably to fixed time traffic signals. These fixed time traffic signals repeatedly respond to a prescribed signal timing program and not to the actual traffic conditions. Hence, research into fixed time traffic signals and their control strategies is an ongoing endeavor. Operating fixed time signals offers different means of controlling traffic. On the one hand, some signal parameters adjustments influence the traffic flow locally at a single junction. In many situations, such a local calibration of the signals turns out to be sufficient to cope with the aforementioned problems. On the other hand, in 1

8 2 Introduction situations with constant high traffic, another non-local control strategy for the fixed time signals becomes more significant: the coordination of the signals. Coordinating traffic signals means the following: coupling of signals via a parameter called offset. This quantity specifies how green phases of different signals are shifted (or offset) to each other. Most prominent coordination objectives are so-called green waves, where vehicles travel without being impeded by a signal showing red. Nevertheless, when considering networks of signals instead of arterials of signals, it is often not possible to adjust green waves for the whole network. Instead, the goal green wave has to be replaced by a more practical term like minimum possible delay. Hereby, the item delay refers to waiting times of vehicles facing red at the signals. Many approaches and models have been proposed in order to find good coordinations of signals in networks. Still, the majority of them reveal shortcomings either way, be it unrealistic modeling of real-world circumstances or the fact that they do not give guarantees for their solution quality. To summarize, there is a need for a mathematical optimization approach for coordinating fixed time traffic signals in networks. This discussion on new required control strategies for fixed time signals is not a theoretical one. Rather, the industry, i.e., traffic companies that plan, manage, and control traffic, demands applicable approaches for coordinating traffic signals in networks. As an indication thereof, we briefly report on an industry project that emerged between the TU Berlin and the PTV AG, which is a traffic planning software company from Karlsruhe, Germany. In this project, the aim was to develop mathematical optimization software to coordinate fixed time traffic signals in networks. During the project, we developed a mixed-integer linear programming approach, which minimizes the delay of vehicles in a network by adjusting optimal offsets. However, several other functionalities were incorporated in the model. The outcome of the project with the PTV, though, is a concrete implementation of the optimization approach, which is about to be included in PTV software soon. In our mixed-integer linear program (MIP) for the coordination of traffic signals, a particular physical constraint has been modeled. This constraint, which we will therefore call Cycle Constraint, has to be formulated for all cycles l C of the graph G that represents the network of signalized junctions. It suffices, however, to state the cycle constraints for the elements of a cycle basis. This then implies these constraints for all cycles of G. Depending on the respective application, though, it has to be a cycle basis with a certain property. In our case of a MIP for coordinating fixed time signals in networks one has to define the cycle constraints for the elements of an integral cycle basis. This means that any integral cycle basis can be used to define the cycle constraints for our MIP. Although any two integral cycle bases lead to MIPs with equal optimal objective value, the computational behavior of their MIPs may be different. Observe that this may be of importance, since we are considering networks of large size where one may not come up with optimal solutions.

9 Introduction 3 A quantity one can use to compare cycle bases is the so-called width of a basis. Loosely speaking, the width of a basis is defined as the product over all cycles of a basis of the number of possible values that the integer variable for the cycle constraint for that cycle can take. Thus, the width of a basis gives an impression of the size of the MIPs feasibility region. The hope is that the smaller the width of a basis, the better the corresponding MIP performs. Among the class of integral cycle bases strictly fundamental cycle bases are a prominent subclass. For a graph G = (V, E), a strictly fundamental cycle basis B is defined by a spanning tree T of G. In particular, the cycles of B are exactly the ones induced by non-tree edges of T with respect to the graph G. Then, in the Minimum Strictly Fundamental Cycle Bases (MSFCB) problem one seeks a spanning tree T that induces a basis of minimum length. In 1982, Deo et al. [DKP82] proved the MSFCB problem to be NP-complete for general graphs. Since then, many heuristics for the MSFCB problem have been proposed. Nevertheless, for comparing the results of these heuristics, i.e., whenever concrete experiments were conducted, sample graph classes were considered. Besides random graphs, grid graphs are the most important such graph class. Grid graphs are also of interest for the two following reasons. First, considering the coordination of traffic signals, many real-world networks have a grid-like structure. One only has to think of the layout of central areas in north american cities. Second, for the MSFCB problem, grid graphs turn out to be computationally tricky. This fact is probably due to an extreme amount of symmetric spanning trees on grids. In 1995, Alon et al. [AKPW95] proved that for square grids with n vertices, the size of an optimal solution to the MSFCB problem is in Θ(nlog n). Still, we decided to investigate bounds on the optimal value of an MSFCB on a square grid having the form c 1 n log 2 n o(nlog n) OPT n c 2 n log 2 n + o(nlog n). We could prove that the above statement is true for c 1 = 1/12 and c 2 = 0.979, respectively. An optimization problem that is closely related to the MSFCB problem is the one of finding a t-tree spanner with minimal t, [CC95]. In this problem, one seeks a spanning tree T for a given general graph G, such that the maximum over all pairs of vertices (u, v) V V \ {(v, v) v V } of the ratio d T (u, v)/d G (u, v) is minimal. Here, d G (u, v) refers to the length of a shortest path between u and v in G. The quantity d T (u, v) denotes the length of the path between u and v in T. The relation between finding a minimal t-tree spanner of a graph and an MS- FCB can be noticed when considering the following unified notation for tree spanner (UNTS) problems. In the UNTS, a problem is defined through a triple (goal, domain, term). Here, goal is either the maximum stretch or the average stretch. Second, as domain, either all non-tree edges or all edges or all pairs of vertices are considered. Finally,

10 4 Introduction term may be one of the following four: d T (u, v) or d T (u, v)/d G (u, v) or d T (u, v)+w(e) or d T (u, v)/w(e), with w(e) denoting a weight of an edge. Although not all combinations of goal, domain and term are possible, there remain 20 tree spanner problems, classified by the UNTS. Interestingly, these 20 notationally different problems collapse to 12 with a general weight function w and to only five, when considering 0/1-weights on the edges. Among these five problems that do not coincide even in the unweighted case, are besides the MSFCB problem prominent optimization problems like the Minimum Average Stretch Spanning Tree Problem [PT01], the Shortest Total Path Length Spanning Tree Problem [DKP82, WCT00] and the Minimum Diameter Spanning Tree Problem [HL02]. Generally speaking, the UNTS provides a classification of related problems, which had not been realized as such before. Hence, interconnections can be revealed and properties like complexity status or inapproximability factors can be carried forward between the problems. The chronology of topics within this introductory part is reflected in the organization of the thesis. Outline of the Thesis In Chapter 1, we investigate the Network Signal Coordination (NSC) problem. After a short introduction of the most important traffic engineering terms related to traffic signals, we first examine a study of related work. In the case of the NSC problem this turns out to be of importance in order to clearly restrain the problem from other optimization tasks regarding traffic signals. Thereafter, we formally define the NSC problem and report on similarities to the related Periodic Event Scheduling Problem (PESP). Moreover, we take advantage of the PESP in order to prove the NSC problem to be NP-complete. Then, in the main part of the chapter, we present a model for the NSC problem. In particular, we develop in detail a mixed-integer linear programming (MIP) approach to solve the NSC problem. We conclude the chapter with a discussion of possible applications in practice of an NSC model in general and the MIP approach in particular. In the mixed-integer linear programming formulation for the NSC problem, the sub-problem of finding appropriate integral cycle bases arises. In Chapter 2, we consider the problem of finding Minimum Strictly Fundamental Cycle Bases (MS- FCB) on grid graphs. In particular, we investigate lower and upper bounds for this problem. As for the lower bounds, we consider both combinatorial approaches and mixed-integer linear programming formulations of the problem that we enrich with several additional cuts. Thereafter, we consider upper bounds for the MSFCB problem on grids. In particular, we construct trees by making intensive use of recursively defined sub-structures. We conclude the chapter with an experimental section in

11 Introduction 5 which we provide benchmark results for the MSFCB problem on grids which help evaluating further research. The MSFCB problem can be interpreted as a problem of finding a spanning tree that minimizes the sum of path lengths between particular pairs of vertices in a given graph. Interestingly, finding minimum average stretch tree spanners or min-max stretch tree spanners of graphs can be interpreted in a very similar way. In Chapter 3, we provide a classification of several problems that aim at finding spanning trees in a graph, which minimize the average or the maximum value of certain distances between particular pairs of vertices in a graph. We propose a unified notation for these problems, which include several prominent problems in combinatorial optimization. With this notation at hand, we identify all coincidences and anti-coincidences of these problems. Moreover, we provide a missing complexity status for one of the problems and observe that an inapproximability result of one of the problems can in fact be applied to another problem too, where it had previously been unknown. In Chapter 4, the experimental work that is related to the Network Signal Coordination (NSC) problem is presented, thereby coming full circle back to the first chapter. The experiments conducted are threefold: First, we perform a solver comparison at some example instances for our MIP model. Here, we compare the MIP solvers CPLEX, MOPS, and SCIP with respect to their ability to find good solutions in short time. In particular, we run two series of experiments using once the default MIP solver settings and once settings that emphasize the finding of good primal solutions. Second, we report on the influence of cycle bases to the computational behavior of the MIP for the NSC problem. This experiment is of general interest, because a positive influence of short bases on computation times of mixed-integer programming formulations of practical applications is expected although very few studies actually proved it. So, in particular, we investigate the correlation between the width of a cycle basis and the lower bound obtained by a MIP computation of 10 seconds. Finally, the third series of experiments is probably the most important one: we evaluate our model by carrying out case studies. Namely, we consider the real-world inner city networks of Portland and Denver and compare the results obtained by our optimization approach with results found by other means. For these comparisons, we use the microsimulation tool VISSIM. How to read this thesis The thesis is chronological in structure. However, the chapters can be followed independently, too. Chapter 2 and Chapter 3 come with their own introduction and consider related, but individually presented, problems. Furthermore, these two chapters do not explicitly require the reading of the Chapters 1 and 4. On the other hand, the Chapters 1 and 4 are strongly related and we recommend that Chapter 1 is read prior to Chapter 4.

12 6 Introduction Moreover, we give a chapter outline at the beginning of each chapter. Also, conclusions are drawn and open questions are raised at the end of each chapter. A further remark We assume the reader of this thesis to be familiar with the basic concepts in linear and integer programming, graph theory, and complexity theory. For additional information on linear and integer programming we refer to [Sch86, NW88]. Good textbooks on graph theory are for example [Wes96] and [Die00]. Moreover, concepts in complexity theory that are necessary to follow this thesis are covered by [GJ79] and [HO02]. As for the parts that deal with traffic engineering concepts or with traffic signal terms in particular, we refer to Section for short textual explanations of the most important terms. Herewith, following most parts of Chapter 1 and Chapter 4 should be unproblematic. Of course, while developing our mixed-linear integer program in Section 1.3, we give formal definitions of all relevant terms, too. However, additional information can be found in Richtlinien für Lichtsignalanlagen RiLSA, Lichtzeichenanlagen für den Straßenverkehr [ril92] and in the Highway Capacity Manual [hcm00].

13 1 THE NETWORK SIGNAL COORDINATION PROBLEM In this chapter, we consider the Network Signal Coordination (NSC) problem. The NSC problem was introduced in 1975 by Gartner et al. [GLG75a], though many similar optimization tasks were known and have been defined already a lot earlier. After a brief summary of the most important terms concerning traffic signals in Section 1.1.1, we give a short overview of the most significant optimization problems on traffic signals in Sec , also to be able to clearly define the NSC problem and to restrict it from other problems. Then in Sec we formally define the NSC problem and illuminate similarities to the Periodic Event Scheduling Problem (PESP) in Section We report on the complexity of the NSC problem in Section Thereafter, in Sec. 1.3 we develop in detail a revised mixed-integer linear programming (MIP) formulation for the NSC problem. Finally, we explain a possible application of our MIP in practice, see Section 1.4. Parts of this chapter were published in [MNW06]. 1.1 Introduction to Coordination of Signals in Networks Before we define the Network Signal Coordination Problem, we introduce the most important terms related to traffic signals and give an overview of what kinds of optimization tasks concerning traffic signals have been considered so far. Other surveys on the topic are provided for example by [SS95, tft] or contained in [Läm07] The Language of Traffic Signals There is no unique language in the field of traffic engineering in general, and neither in topics related to traffic signals. Rather, the terms and notation depend on the 7

14 8 The Network Signal Coordination Problem respective country and language. However, since following this thesis requires only basic knowledge of traffic terms, in this section we give only a short textual description of the most important terms. Notice that we do not give formal definitions here. For those, we refer to section 1.3. Nevertheless, whenever it is possible, i.e., when we do not work with a term, but only want to give an intuition, we omit a formal definition at all. For complete information see [hcm00] and [ril92]. In traffic engineering one considers traffic, i.e., vehicles, moving through a single and isolated junction, an arterial, which is a, possibly bi-directionally traversable, series of junctions, or through a whole network, i.e., an arbitrary set of junctions. Then, one distinguishes different types of signals at the junctions. Here, the term signal refers to all signaling devices at a junction. Roughly speaking, the following two types are the most important ones. On the one hand, there are traffic responsive signals. At these signals, the signal settings react on the present traffic. On the other hand, fixed-time (controlled) signals do not react on the actual traffic. Here, after a prescribed time span, called cycle length, the pattern of red phase and green phase repeats. The particular division of a cycle length into a red and a green is referred to as (red green) split. At a fixed-time signal, there are usually different signal groups that control the traffic for particular directions. The green phases of different signal groups are shifted against each other, since they usually control competing traffic streams. In addition, the order of the signal groups at a signal is called phase sequencing. See Fig. 1.3 on page 20 for an example of a signal timing plan in which the relevant data for one fixed-time traffic signal is merged. A very important term is the one of an offset. The (inter node) offset determines how different signals are operated or shifted relatively to each other. That means the following: at each signal there is a marked out reference point, which sometimes is the begin of the green phase of the first signal group. Then, the offset denotes the time span between reference points of two signals at two consecutive junctions. In this case, there is an offset for each pair of consecutive signals. However, the offset can also be defined for one single signal. Then, it determines the time span between this signal s reference point and a given network-wide zero reference point. See Figure 1.4 for an illustration of both types of offsets. A sketch of the intra-node offset, which determines the shifting of different signal groups at one signal, is depicted in Figure 1.5 for example. Of course, when considering signalized junctions, arterials or networks, several optimization tasks come to mind. Generally, one is interested in optimizing signal settings in order to achieve a certain goal. Such signal settings are the red green split, the phase sequencing, the cycle length, and the offset. As for the goals to achieve, for example, minimizing the delay or maximizing the bandwidth have to be mentioned. Here, the term delay refers to the delay that is due to the signalization, i.e., delay that occurs when vehicles have to wait because of a red. On the other hand, maximizing bandwidth means that the signals along an arterial or within a network are adjusted, such that a preferably wide possible corridor through the green phases of consecutive signals exists, within which the vehicles do not have to stop at the signals at all. Such a corridor is sometimes called a greenband. See Figure 1.1

15 1.1 Introduction to Coordination of Signals in Networks 9 for a visualization of greenbands. Whenever the offsets are included in the signal settings to be optimized, we say that we optimize the coordination. In the literature the term synchronization is sometimes used synonymously. However, we prefer the term coordination and leave the item synchronization to cases where the offsets and the cycle length are optimized. When considering traffic flow one distinguishes between a microscopic view and a macroscopic view. The model is said to be microscopic if each individual vehicle is considered. On the contrary, we speak of a macroscopic model or approach, if the vehicles are aggregated in some sense. For example, it is popular to consider platoons of vehicles, that is, groups of consecutive vehicles close together that are treated as one quantity. However, it has to be mentioned that there are traffic models for which a classification into one of the two views is not obvious Optimizing Traffic Lights Since the introduction of automatic traffic signals in the 1920s, much work and research has been done on modeling, analyzing, and later also on simulating and optimizing traffic signals. In this section we mention the most important modeling and optimization approaches. Notice, however, that we do not claim to provide a complete overview. When talking about optimization in the context of traffic signals, one faces many different optimization tasks. Table 1.1 gives a glimpse of possible differentiations between them. One of the first important scientific publications on traffic signals was by Webster [Web58] in In this pioneering work, he prepared the ground for analyzing single traffic signals, e.g., by providing delay-estimating formulae that are, in a slightly changed form, still in use today. Using this formulae, Webster also researched on minimizing the delay by adjusting optimal green proportions at a signal. Then, during the 1960s research was no more restricted to one single junction, but rather arterials and networks of signals were considered. In 1963, Newell [New64] investigated the coordination of signals with certain assumptions on the density of traffic along an arterial of one-directional traffic. In his macroscopic model he suggests that best coordination is achieved simply by coordinating two consecutive signals at a time. Morgan and Little [ML64] and Little [Lit66] then developed an optimization model that maximizes bandwidth, for the first time using mixed-integer linear programming. In their approach, the authors adjust optimal values to offsets, a common signal cycle length, and progression speeds, considering a bi-directional arterial of signals. In the mid-1960s a first big field-study in coordinating a real traffic network was carried out in the city of Glasgow, Scotland. In this experiment, conducted by Hillier [Hil65, Hil66], different types of signal controllings were tested in inner city sub-network in Glasgow to evaluate their benefit. Later, in 1967, Hillier and

16 10 The Network Signal Coordination Problem Table 1.1: The table provides criteria to distinguish between mathematical approaches for problems dealing with traffic signals. Observe, however, that not all combinations are reasonable. Criteria type of approach variables objective type of signal type of approach application on preconditions on traffic preconditions on demand preconditions on signalization modeling perspective Possibilities optimization, heuristic (genetic algorithms, local search etc.) offset, red-green split, cycle length, phase sequencing, travel speed, routes, almost any combination thereof minimizing delay, number of stops, fuel consumption; maximizing greenband; combinations thereof fixed-time signals, traffic responsive signals, both theoretical, practical single junctions, arterials, networks public only, individual only, none high demand only, low demand only, none common cycle length, none macroscopic, microscopic Rothery [HR67] analyzed the relation between platooning behavior of vehicles and coordination. In particular, they calculate total delay as a function of the offsets. For this study the authors investigate four signalized junctions in London, England. At the same time, a graph theoretical model for optimizing the coordination in order to minimize the delay was developed by Allsop [All68]. In his article he proposed an iterative approach that successively expands the considered sub network for which a solution had already been found. In 1969 the theoretical work for one of the until now most widely used software tools for the optimization of coordination was published by Robertson [Rob69]. We refer to the next section for a more detailed discussion of the properties and attributes of TRANSYT. Then, in a series of publications Gartner [Gar72] and Gartner et al. [GLG75a, GLG75b, GLG76] developed an mixed-integer linear programming approach for network coordination. In fact, the optimization model that we introduce in Section 1.3 is based on their approach. At the same time, another approach evaluated the possibilities of linear programming for optimizing traffic signal settings. Antoniadis [Ant75], though, did not include the offset into his model. A next, also from the theoretic point of view, important approach was the one by Improta and Sforza [IS82] in There, the authors developed a mixed-integer linear program, similar to the one by Gartner et al. [GLG75a], but included a branch

17 1.1 Introduction to Coordination of Signals in Networks 11 and backtrack method that relaxed certain assumptions on the delay functions, which had been made by Gartner et al. Dauscha et al. [DMN85], then introduce in 1985 a very general problem framework for finding cyclic schedules of task systems. Nevertheless, their problem formulation is strongly motivated by coordinating traffic signals. In addition, they are the first who report on the complexity of scheduling periodic events or coordinating signals, respectively. In 1989, Serafini and Ukovich provide two important contributions. In [SU89b], they develop an involved mathematical model specially tailored for the fixed-time traffic control problem, i.e., for the coordination of fixed-time signals. Second, in [SU89a] they present a widely noticed general approach that schedules periodic events, which is, though, very similar to [DMN85]. Then in 1991, a second computer-aided approach, named SCOOT 1, was published by Robertson and Bretherton [RB91]. However, in contrast to TRANSYT, SCOOT applies to adaptive traffic signals. Later, in 1996 Hassin [Has96] presents a flow algorithm approach to the synchronization of networks with an explicit application to the synchronization of fixed-time traffic signals. The approach consists of a local search heuristic and, moreover, a characterization of local optima is given. In addition, comparisons to other heuristic approaches like TRANSYT were carried out on a network in the city of Tel Aviv, Israel. A heuristic optimization approach that bases on genetic algorithms was proposed by Almasri and Friedrich [AF05] in There, the authors use a cell transmission model which was originally introduced by Daganzo [Dag95]. However, only adaptively controlled signals are considered. In 2005, Braun and Weichenmeier [BW05] introduce a second heuristic approach. In their model, the authors consider several signal settings, including offsets, and propose a genetic algorithm method. Nevertheless, they report on test runs of the approach on networks of small sizes only. In order to show that modeling and optimizing traffic signals indeed attracted researchers from different fields of expertise we give the following two citations, too: in 1994 Ianigro [Ian94] developed a traffic model by using petri nets. Then, he used this model to implement a simulation that finds optimal signal settings for the considered traffic network. A second interesting approach is the one by Gershenson [Ger05]. In 2005 he considered traffic flow in a signalized network in the context of self-organizing systems, i.e., he investigates self-organizing traffic lights that adapt to changing conditions. Other important contributions that we did not mention in detail before are [Sto68, CI88]. For an broader overview we refer the reader to [KN03]. 1 Split Cycle Offset Optimisation Technique

18 12 The Network Signal Coordination Problem Software to Optimize Coordination Whereas the last section was intended to give an overview of optimization approaches to traffic signal related tasks, this section explicitly reflects some software packages that optimize the coordination of fixed-time signals, both on arterials and within networks. However, again, we do not claim completeness, but rather mention the to our knowledge three most important ones: TRANSYT [Rob69], MITROP [GLG76] and SYNCHRO [syn00]. Remark 1.1. Though nowadays, coordinating traffic signals is done mostly computer aided, still, some techniques are in use, where computers only provide graphical support. For example, especially for arterials, coordinating via a graphic-based by-hand approach that visualizes greenbands in dependence of offsets is common, see Figure 1.1. Figure 1.1: A still today quite usual method to determine a good coordination on arterials: a graphical approach 3 by hand. Notice the greenbands in green and blue for opposite traffic. Unquestionably, the software tool TRANSYT 4 is state-of-the-art with respect to relevance for practitioners who want to model, analyze, and optimize traffic signal settings for a network. The wide acceptance of TRANSYT as well as its importance is stressed by the following quotation. The TRANSYT method serves as an unofficial international standard against which to measure the efficiency of other methods of coordinating networks of traffic signals. [RB91] The software tool TRANSYT is based on an approach by Robertson [Rob69], published in In this approach, fixed-time signals are considered and signal settings are improved via a gradient (hill climb) search technique or a genetic algorithm approach. Namely, a so-called performance index (PI) is minimized. An advantage of TRANSYT is the very detailed objective function, i.e., many aspects can be taken 3 By courtesy of the PTV AG. 4 The acronym TRANSYT stands for TRAffic Network StudY Tool

19 1.2 The Network Signal Coordination (NSC) Problem 13 into account for the PI. Since in Section 4.3 we work with TRANSYT, we now give some detailed information on its mode of operation and the adjusted parameter setting in the following remark. Remark 1.2. We use version Transyt-7F 10.1 and its genetic algorithm approach to optimize offsets in a network. The genetic algorithm parameters are: crossover probability = 40%, mutation probability = 2%, convergence threshold = 0.01%, number of generations = 100, and population size = 25 which are the defaults. Further, we calibrated the performance index PI such that it measures delay only. As the mode we used both the single-cycle mode. To summarize this paragraph: when developing a model that optimizes the coordination of traffic signals in a network, evaluating it by comparing the results to the ones of TRANSYT is a must. In 1976 Gartner, Little and Gabbay introduced the software tool MITROP 5 that optimizes traffic signal settings [GLG76, GLG75a, GLG75b]. By then, the MITROP was one of the first approaches that used integer programming techniques. In addition, Gartner et al. s model was the first one to simultaneously optimize (one network wide) cycle length, red-green splits at the signal and offsets. However, since the relation between these signal settings is quadratic, they use piece-wise linear approximations. Our approach, which we present in Section 1.3 is actually based on MITROP. Hence, we adopt most of their notation. Nevertheless, shortcomings of the MITROP model are the objective function, see Section for more information, and the assumption of a global cycle length. However, to the best of our knowledge, MITROP has not had commercial success, thus, it may be considered a theoretical approach. A third software tool, which has to be mentioned when speaking of coordination of signals in a network, is SYNCHRO [syn00]. Very similar to TRANSYT, the tool SYNCHRO models all traffic signal settings and searches heuristically for a solution with small average delay. However, the three briefly introduced software tools, TRANSYT, MITROP, and SYNCHRO are by far not the only computer programs that deal with the coordination of signals in networks. Still, we consider them to be the most relevant ones to compare our approach with. 1.2 The Network Signal Coordination (NSC) Problem In this section we consider the Network Signal Coordination (NSC) problem. First, in Section we give a definition of the problem. Thereafter, in Section 1.2.2, we classify the NSC problem to related problems like the PESP that also deal with periodically repeated events. Finally, in Section we prove the NP-completeness for the NSC problem. 5 MITROP abbreviates Mixed Integer Traffic Optimization

20 14 The Network Signal Coordination Problem Definition of the NSC In this section we will give a mathematical problem formulation for the Network Signal Coordination (NSC) problem. However, the problem formulation will be as what we want to be understood by network signal coordination. Obviously, there is no unique possibility to define the problem mathematically. Nevertheless, none of the already known problem definitions suffices our needs. Either, they are formulated too general, or, they are formulated too narrow, meaning that many specialized assumptions regarding the signal or the traffic flow, e.g., platoon lengths, are incorporated into the problem formulation. So, we try to find a problem formulation that is tailored to the coordination of traffic signals in networks, while not being over-restrictive with special assumptions, e.g., on lengths of platoons. Still, we want our problem formulation to map the following phenomena that are inherent to the practical problem. In what we refer to as the practical problem we are considering the following setting. We are given a network of junctions which are signalized with fixed-time traffic signals. The signals in the network have uniform cycle length. Then, the objective of the practical problem is to adjust offsets at the signals such that delay that is due to poor coordination, is minimized. Moreover, we want our problem formulation to map a macroscopic traffic scenario. Obviously, this is, in general, a quite restrictive assumption. Nonetheless, under certain conditions, on which we report in Section 1.3, the macroscopic view is accepted. Then, as a consequence, we assume the objective function to be separable. This means that for one link, the delay of the vehicles only depends on the offsets of the two incident signals. Hence, we may consider each link individually, i.e., the objective is a sum of functions that evaluate the link performance: so-called link performance functions (LPFs). Finally, we think it is both, sufficiently exact and sufficiently general, to assume the LPFs to be continuous and piece-wise linear. Mathematically, we model the network as a directed graph D = (V, A), where we refer to an element v V synonymously as junction, signal, node, or vertex. An element a A we call an arc. We use link or edge synonymously. The network wide cycle length of the signals is denoted by c with c É and the vector π É V is the offset vector for the signals. First, the following observation can be made. In the above requirements, we did not mention any constraint on an offset π v, v V. Hence, any π É V constitutes a feasible solution, or can be turned into a feasible solution, respectively, by replacing π v with π v mod c. Thus, formulating the network signal coordination problem as a decision problem, which answers the question whether there is a feasible set of signal offsets does not make any sense. Rather, we consider the network signal coordination as an optimization problem. However, we formulate the optimization problem by its corresponding decision problem using a parameter K. We define the Network Signal Coordination (c-nsc) problem as follows.

21 1.2 The Network Signal Coordination (NSC) Problem 15 Network Signal Coordination (c-nsc) Problem Instance: Task: A directed multigraph D = (V, A). Continuous, piece-wise linear functions h a : É É + for all a A that fulfill h a (x) = h a (x+c) for all x É. A number K É +. Find a vector π É V, such that h a (π v π u ) K a=(u,v) A or decide that no such vector exists. We refer to the problem version where the cycle length c is part of the problem input as NSC. The following has to be noticed. We define the NSC problem using a network wide uniform cycle length c. Actually, for this case one can restrict the functions h a a little more without losing practical relevance. Namely, one can claim piece-wise convexity for the h a. In particular, we assume that it holds that for all a A there exists an x É such that h a is convex on the interval [x, x+c]. Nevertheless, since we are going to consider the NSC problem with non-uniform cycle lengths, too, we omit the convexity assumption on the functions h a, because in this case, the piece-wise convexity property is hard to reconcile with observations from practice. Notice that our above definition of the c-nsc is very similar to other problem definitions. For example, Hassin [Has96] introduced a problem, which he called network synchronization problem. There, he also optimizes a separable function of differences of node potentials. However, for the problem formulation, Hassin does not make any assumptions on the type of functions, although, when developing his approach, he also reports on periodic, continuous piece-wise linear function. Still, for our definition of the c-nsc, we decided to include assumptions on the type of the objective as well as the periodicity. We think that the periodicity and the continuous piece-wise linear functions characterize well a general network signal coordination problem The Relation Between the NSC and the PESP This section is dedicated to illuminate coincidences between the NSC problem and related problems such as the Feasible Differential Problem (FDP) and the Periodic Event Scheduling Problem (PESP). We are aware that the problem formulation of the NSC problem overlaps with many known problems. One major difficulty to properly classify the NSC problem is that different streams of research dealt with very similar problems. For example, traffic engineers, as summarized in Section 1.1.2, consider mostly motivated from practice problems concerning signal coordination, which are formulated very close to the NSC problem. Nevertheless, there was no uniform mathematical problem formulation established. Other decisive influences came from the field of operations research and from computer scientists. In 1985, Dauscha et al. [DMN85] proposed

22 16 The Network Signal Coordination Problem an approach for cyclic schedules for task systems where they put the problem of coordinating signals in a much more general framework. In addition, it was this article that reported for the first time on the complexity of a problem closely related to coordinating signals. However, it was not until 1989 that Serafini and Ukovich proposed a problem formulation and a notation that became broadly accepted. In particular, in [SU89b] they introduced an approach specialized for coordinating signals in networks. Moreover, Serafini and Ukovich published another article in the same year. In [SU89a] they introduce a mathematical model for periodic scheduling problems. Although their problem definition is very similar to the one in [DMN85], their formulation of a Periodic Event Scheduling Problem (PESP) attracted much more attention. The PESP can be seen as the periodic extension of the so-called Feasible Differential Problem (FDP). Feasible Differential Problem (FDP) Instance: A directed graph D = (V, A) and vectors l and u É A. Task: Find a vector π É V such that for every arc a = (v, w) it holds l a π w π v u a or decide that no such vector exists. In both problems, the FDP and the PESP, one considers a directed graph D = (V, A) and seeks a vector π É V. This setting is equal to the one for the NSC problem. However, for the FDP and the PESP, one additionally is given vectors l and u É A. With these vectors, constraints on π are formulated, non-periodic ones for the FDP and periodic ones for the PESP. Notice that the absence of such constraints is the major difference between the NSC problem and these two problems. However, it is a trivial observation that the c-nsc problem with K = is equivalent to the PESP with l a = 0 and u a = T for all arcs a A. Periodic Event Scheduling Problem (T-PESP) Instance: A directed graph D = (V, A) and vectors l and u É A. Task: Find a vector π É V such that for every arc a = (v, w) it holds l a (π w π v ) mod T u a or decide that no such vector exists. When comparing the NSC to the PESP, the following facts have to be mentioned. Because of the observation that the feasibility version of the NSC is contained in the PESP, one could think that the NSC problem is simply a special case of the PESP. However, this is not true, as can be seen when considering the optimization problem versions of the two problems. In particular, we consider the optimization problem version of the PESP as it is used for its main application: periodic timetabling. Then, on the one hand, one assumes separable objectives for both problems. On the other

23 1.2 The Network Signal Coordination (NSC) Problem 17 hand, different types of functions are considered to measure the link performance for the two problems. For the NSC problem, we assume the performance function for a link to be continuous and piece-wise linear. When optimizing periodic timetables with the PESP, though, often non-continuous functions that are periodically linear on an interval of length T, are considered. Moreover, due to the different origins, different names for the model quantities have become standard. The period length T of the PESP, for example, is equivalent to the cycle length c for the NSC problem. Moreover, in the PESP context, the π É V, are called potentials, whereas when coordinating signals, they are called node offsets. To summarize, the NSC problem and the PESP can be regarded as quite similar. The differences lie in the bounds on the vector π for the PESP and in the different objective functions. Therefore, modeling or solution approaches to the two problems, as for example MIP formulations, differ as well, little, but noticeable. However, we do not go into further detail for that question. Another interesting aspect when comparing the NSC problem to similar problems does not become apparent until considering a special model for NSC. It is possible to model the NSC problem not using variables π v for a v V, but exclusively with variables φ a := π v π u for each a = (u, v) A. When doing so, one has to add certain constraints to the model in order to be able to recalculate the node offset π v out of the link offset φ a of a solution. 6 An important observation is that such constraints do also appear in the according model for the PESP and, generally, in every approach that models periodically repeated events using variables on arcs instead of variables on nodes. Often, these constraints are called cycle constraints, but, actually, they have had various names like congruence equations [SU89b] or loop constraints [GLG75a]. In fact, these cycle constraints were already formulated by Kirchhoff [Kir47] in an aperiodic manner, i.e., for describing properties of voltages in electrical networks. Finally, we give a short résumé on this section. In the 1950s and 1960s traffic engineers developed the first models for coordinating traffic signals in networks. Although by then the problems were approached more from the practical side, many sophisticated mathematical solution methods were proposed. Then, during the years, the problem of coordinating signals inspires researches from operations research and computer science. First, they come up with own approaches to signal coordination and provide first complexity studies. Thereafter, they develop much more general problem frameworks that cover many different real-world applications like railway timetabling [Lie06]. Nevertheless, a backfertilization to the actual origin, i.e., the coordination of traffic signals, could be observed. For example, new mathematical expertise in network flow theory [Has96] entered the signal coordination approaches. 6 See Section for a detailed discussion of these constraints at the example of modeling the coordination of traffic signals.

24 18 The Network Signal Coordination Problem NP-completeness of the c-nsc problem In 1975, Gartner et al. [GLG75a] introduced an optimization problem that minimizes the delay of vehicles in a network with fixed-time signals by adjusting optimal offsets. However, although providing a mixed-integer approach for their problem, which they called Network Coordination Problem, Gartner et al. did not report on the complexity of the problem. In 1985, Dauscha et al. [DMN85] proved the NP-completeness of their Cyclic Schedule Problem, which can be considered to be very close to the problem of coordinating traffic signals. They reduced the Graph k-colourability Problem, see [GJ79], to their problem. Later on, other hardness proofs for slightly more general, problems followed. In 1989, Serafini and Ukovich [SU89a] provided an NP-completeness proof for the PESP. They reduced the Hamiltonian Circuit Problem, see [GJ79], to their problem. In 1996 Hassin [Has96] proves the NP-completeness of his Network synchronization approach by a reduction from the Minimum Cluster Problem, see [GJ79]. The c-nsc problem of Section obviously belongs to the class NP. Hassin [Has96] reported on the NP-completeness for a variant of his problem which implies the hardness of the c-nsc problem. Still, in Theorem 1.3 we provide an alternative reduction from T-PESP proving the c-nsc problem to be NP-complete for c 3. Theorem 1.3. The c-nsc problem is NP-complete for c 3. Proof. It is obvious that the c-nsc problem belongs to NP. We show the NP-hardness by a reduction from T-PESP, see Section for a definition, which is known to be NP-complete for T 3 [Odi94, Odi96]. Let I = (G = (V, A), l, u É A ) be an instance of T-PESP for some T 3. Define an instance I = (G = (V, A ), h a, K) for the c-nsc problem in the following way. First, let G := G, i.e., we take the graph G with its vertex set V and its edge set A. Further, we set K = 0 and c = T. The functions h a are defined as follows. For an a A we set 2 c (u a l a ) t + h a (t) = 0, if t [l a, u a ], 2 c (u a l a ) t 2 u a c (u a l a ), if t [l a c (u a l a ) 2, l a ], 2 u a c (u a l a ), if t [u a, u a + c (u a l a ) 2 ] which defines h a on the interval [ I := l a c (u a l a ), u a + c (u ] a l a ) 2 2 (1.1) which has length c. For t / I, we set h a (t) = h a (t ) with t = (t mod c) + l a c (u a l a ) 2, where we observe that t I. Notice that with this definition, the functions h a are continuous, piece-wise linear, and fulfill h a (x) = h a (x+c) for all x É. The definition of the h a is motivated as follows. We can assume without loss of generality that for the PESP instance I, it holds for all a A that u a l a < T.

25 1.3 A Revised MIP Formulation for the NSC problem 19 Moreover, the functions h a are defined such that h a (t) = 0 if and only if 0 t mod c u a l a. See Figure 1.2 for an illustration of one function h a. c h 1 a (t) l a u a t Figure 1.2: We deduce a function h a using the PESP bounds l a and u a, such that h a (t) = 0 if and only if 0 t mod c u a l a. However, if the instance I is a Yes-instance, then there exists a vector π É V such that h a (π w π v ) = 0. a=(u,v) A Thus, h a (π w π v ) = 0 for all a A. Hence, 0 (π w π v ) mod c u a l a, which means that I was a Yes-instance for the PESP. It is obvious from the construction that the other direction works out the same, i.e., that a Yes-instance I for the PESP is always recognized by a corresponding Yes-instance I of the NSC problem. Notice, that an APX-hardness proof from the optimization version of the T- PESP see [Lie05, Nac96a] does not carry over to the c-nsc problem. The reason for this is that the proof makes use of the discontinuity points in the objective of the T-PESP; in the c-nsc problem we have a continuous objective function. 1.3 A Revised MIP Formulation for the NSC problem In the previous sections of this chapter, we reviewed existing approaches for the coordination of traffic signals in networks, and spent some effort in appropriately defining the Network Signal Coordination (NSC) problem. Now, in this section, a mixed-integer linear programming approach for the NSC problem is presented. In this approach, which is based on an approach by Gartner et al. [GLG76], also [GLG75a] and [GLG75b], we minimize the total delay occurring in a traffic network by adjusting optimal offsets. In fact, in our MIP model, we also allow a variable phase sequencing and a non-uniform cycle length at the signals of the network. Therefore, the MIP approach that we develop in this section actually solves a more general problem than the c-nsc. The section is organized as follows: first, in we introduce the traffic scenario that we consider and discuss the assumptions that we make. Moreover, we introduce the mathematical notation that we need. Note that we adopt most of the notation from Gartner et al., see [GLG75a]. Thereafter, in Sections to we

26 20 The Network Signal Coordination Problem develop the MIP, which we finally state and discuss in Section Notice that in Section we also motivate the major model technique of using link-wise defined offsets instead of node-wise defined ones Introduction We assume the following scenario. We consider an (inner-city) traffic network with fixed-time signals at the junctions. Through that network, vehicles move on prescribed paths. Notice that we only consider individual traffic. Further, non-uniform cycle lengths at the signals are explicitly permitted. Moreover, the network is assumed to operate at near-saturated condition. That means that we mainly face high traffic volumes on the links. Thus, as already motivated by Wormleighton [Wor65], it is justified to model the traffic flow through the network macroscopically, i.e., we do not consider each vehicle individually, but rather consider groups of vehicles, socalled platoons. Doing so, we further assume the traffic volumes to be given link-wise. In the model, all considerations, e.g., the calculation of delay, are done separately for each link. So, we set up a mathematical model that minimizes the sum of delays on all links in the network with offsets between the signals and, what we call split mode or phase sequencing, as decision variables. The traffic network is modeled by a directed multi-graph G = (V, A), where the vertices v V represent the signalized junctions and the edges a A stand for the traffic flow between the junctions. We will also use the terms node and link or arc, respectively. For an edge a = (u, v) A we call the node u upstream node of edge a, whereas v is referred to as downstream node of a. The reason for allowing parallel edges is that we distinguish the traffic flow on a link depending on the signal groups of the upstream and the downstream junctions. Moreover, let L denote the set of circuits of the underlying undirected graph of G. In the following, we use the terms circuit and cycle synonymously. In the remainder, whenever it is obvious from the context we omit indices for different copies of a link. In Fig. 1.3 all relevant data for one signal and their notation, respectively, is depicted. signal groups Figure 1.3: A signal timing plan of Signal v is shown. Exemplarily, the length g of the green phase for signal group 2, and the length r of the red phase for signal group 1 are depicted. c g r v A first remark is that we do not consider amber phases at the signals. Instead, we assume the signal timing plan to only include green and red phases for all signal

27 1.3 A Revised MIP Formulation for the NSC problem 21 groups at a signal. This is no limitation, since it is accepted to transform a usual timing plan with the amber phase into one without an amber phase, using the concept of an effective green. This is carried out in more detail in [GLG75a]. During the following sections, we successively increase the level of detail that we use for explaining our approach. Therefore, for example, the non-uniform cycle lengths are introduced not until Section Thus, assume for now that the signals in the network operate with the same cycle length. Let c É denote the cycle length at the fixed-time controlled signals in the network. Further, the length of a red phase is denoted by r, whereas g stands for the length of a green phase. The indexing for these quantities works as follows: because all considerations within the model are done edge-wise, we index g and r edge-wise, too. That means, for an edge a A, g a denotes the length of the green phase at the downstream signal of a. The length of a red phase r a is defined analogously. Moreover, let τ a denote the travel time on link a, i.e., the time which needs the platoon to move between the two signals that are incident to arc a. Observe, the following notationally simplification: during the next sections, we often refer to an edge via the identifier a A. Sometimes, it will be necessary to refer to the nodes incident to a. Then, we use, e.g., the notation a = (u, v) A. Since we allow parallel edges in the graph, an edge is not sufficiently characterized by its incident nodes. Rather, one actually would have to include the respective signalgroups at the signals u and v, which then uniquely characterize the traffic flow, represented by a. For clarity reasons, and when there is no confusion possible, we thus omit the identifiers for the signalgroups at u and v. To summarize this short introduction: first, we assume the vehicles in the network to move in platoons, and, second, we conduct all modeling steps link-wise The Offsets This section is intended to treat two important tasks. First, we formally introduce the term offset, in both of its possible interpretations. Second, we motivate, the decision for the link-wise defined offsets for our mixed-integer linear program. We begin the development of our mixed-integer linear program with the most important variables, the offsets. Although in the problem formulation for the NSC one is interested in offsets for the signals, there are actually two possibilities to define the offset: node-wise or link-wise. First, we illustrate the difference between the two viewpoints and, second, we give formal definitions. In Figure 1.4 the two possibilities to interpret the term offset are shown. First, in Fig. 1.4(a), we illustrate node-wise defined offsets. There, the term offsets refers to the time span between the zero point in the signal s timing plan and the network s global zero point. These two reference points can be set arbitrary, however, they have to be fixed beforehand. Second, in Fig. 1.4(b), we illustrate a link-wise defined offset. There, the term offset refers to the time span between the beginnings of the green phases of the upstream and downstream signal of the link. Of course, in both cases the offset value is only unique modulo the cycle length c.

28 22 The Network Signal Coordination Problem π network zero point π u (a) node offset signal zero point v u φ (b) link offset Figure 1.4: The two possible interpretations of the term offset are shown. In (a), the offset is defined node-wise, whereas in (b) it is defined for each edge a A. In the example, the link a represents traffic flow between signal group 2 of signal u and signal group 3 of signal v. v For a node v V we denote with π v the node offset at junction v. Further, we require π v [0, c). For an arc a, the link offset variable is denoted by φ a. Later on in the section, we discuss bounds for the offsets φ a. Now, before we formally show the connection between the π v and the φ a, we have to introduce the intra-node offsets denoted by Ψ. For a node v V, let R v be the set of its signalgroups. Then, the intra-node offset Ψ p v refers to the time span between the beginning of the green phase of signal group p R v and the signal reference point to which the node offsets are linked. See Figure 1.7 on page 25 for examples. For now, we assume the intra-node offsets to be an input parameter. Later, in Section they become variables, though. At this point, we can formalize the connection between the node offsets π and the link offsets φ. Consider the edge a = (u, v) A. Assume that φ a [r a +τ a c, r a +τ a ]. This will be motivated later. Then, let os(a) denote the signal group at node u, the origin of a. Further, ds(a) denotes the signal group at the destination of a, the node v. Here, os(a) R u and ds(a) R v. Then, there exists an integer p a, such that ( ) π u + Ψ os(a) u ( π v + Ψ ds(a) v ) + p a c = φ a. (1.2) The connection stated in Eq. (1.2) is rather easy to see. We illustrate an example in Figure 1.5. Hence, the following can be observed. A model that exclusively contains node offsets π can be considered equivalent to a model that exclusively contains link offsets φ. Therefore see Equation (1.2). On the one hand, with fixed values of π u and π v there always exists a unique p a such that a φ a with φ a [r a + τ a c, r a + τ a ] exists. On the other hand, one can uniquely determine the π vector out of the link offsets. However, therefore one node offset has to fixed, which is, though, not a limitation. This is, because two solutions π, π É V can be considered equivalent if one is just

29 1.3 A Revised MIP Formulation for the NSC problem 23 π v Ψ 2 v v π u Ψ 3 u φ a Figure 1.5: The connection between node-wise defined offsets and link-wise defined ones allows for a straightforward translation between the two terms. u a translation of the other, namely, if there exists a constant q, such that π a = (π a + q) mod c, a A. (1.3) However, then, with one π v, v V, fixed, one simply uses (1.2) to uniquely determine all π v propagating the φ a, e.g., along a spanning tree of the underlying graph. That this propagation works out is ensured by so-called cycle constraints, which are discussed in detail in Section Now that we know that there exist several possibilities to model the quantity offset for our signal coordination approach, we discuss why we decided to exclusively include the link offsets φ into our model. The main reason, why we build up the model with the link offsets, is the objective function. Using only the node offsets π v for the model, we would have faced the following problem. As we already mentioned, the objective function is separable. That means, we have an objective function for each arc a A measuring the link performance. These functions have to be defined in dependence of the difference π v π u, for a (u, v) A. However, as it will become clear in Section 1.3.5, where we discuss the objective function in detail, we need to model the objective, such that it is convex on an interval of length c, which we know in advance. Using exclusively the π vector, ensuring such a property, in fact, costs a modulo operation per link, i.e., number of edges many integer variables. As we will see in Section 1.3.3, building up the model with the link offsets only causes ν additional constraint and integer variables, where ν is the cyclomatic number of the graph and ν = A V +1. Still, we will remark on the link offset versus node offset question again in Sections 1.5 and 4.4. So, we decided to use the link-wise defined offset for our MIP approach. However, we introduce a further variable the platoon s arrival time γ which is, though, linear dependent from the offset φ. On link a = (u, v) A, the arrival time of the head of the platoon at junction v is denoted by γ a. The relation between the arrival time γ a and the link s offset φ a, is given by τ a γ a + r a = φ a, (1.4)

30 24 The Network Signal Coordination Problem with τ a being the travel time of the platoon and r a denoting the length of the red phase at the downstream signalgroup of the link. In addition, we require the arrival time γ a to lie in the interval [0, c). This will be useful when we use the γ a for the objective function, i.e., to evaluate the delay on link a. Note that the bounds for γ a imply that for the according link offset, it holds that φ a [r a + τ a c, r a + τ a ]. (1.5) In Figure 1.6 we illustrate that connection between the arrival time and the link offset. r c γ φ τ Figure 1.6: The figure illustrates the simple linear dependency between the arrival time of a platoon, γ, and the link offset φ The Cycle Equations Consider the set L of cycles of G. Note that since we are considering a multigraph, L includes cycles that contain only two parallel or anti-parallel edges. For a particular circuit l L that is traversed clockwise, F(l) is the set of forward edges and R(l) the set of reverse edges, respectively. With c denoting the network s cycle length we require the variables to fulfill φ a φ a = n l c, (1.6) a F(l) a R(l) for all cycles l L and with n l. Then the following connection between (1.2) and (1.6) can observed: Lemma 1.4. For a link offset vector φ É A, the following two statements are equivalent: (i) there exists a vector π É V s.t. (1.2) holds for all edges a A, (ii) for all l L, (1.6) holds. Proof. For simplicity, we assume that all signals have exactly one signal group. Thus, (1.2), reads as φ a = π v π u + p a c. Then, assuming (i), summing up

31 1.3 A Revised MIP Formulation for the NSC problem 25 offsets φ a along a cycle l L results in a telescope sum and (ii) holds. On the other hand, with (ii), one can easily find a π É V fulfilling (1.2): first, fix one π v, for a v V, to an arbitrary value. Then, second, propagate the link offsets along the edges of a spanning tree simply using (1.2). Then, the node offsets π are well defined because (ii) holds. So, the periodicities expressed in Equations (1.6) are physical constraints that are necessary and sufficient to equivalently model the NSC problem with link offsets, instead of node offsets. For example when there is a cycle for which (1.6) does not hold, the transformation of the offsets is not well defined. That means, propagating the φ a using Equations (1.2) brings a contradiction. See Fig. 1.7 for a small example. Note that the intra-node offsets have to be included as well, since the link offsets π A = 65 A π A = 0 φ AB = 35 φ CA = 55? C B φ BC = 55 π C = 10 π B = 35 Figure 1.7: Consider this small example network with a cycle length of c = 80s. Assume that the node offset at vertex A was set to 0. Then propagating the link offset along the cycle reveals node offsets of 35 and 10 for the vertices B and C, respectively. However, if then the link offset on link (C, A) is not equal to but instead, e.g., equal to 55, the node offset at A ceases to be well defined. do not respect changes of signal groups at a junction. This means the following. Suppose, we are considering a circuit that contains two links that share a node v. Further, assume these edges have the same orientation with respect to the clockwise traversal of the circuit. If then the edges do not share the same signal group at node v, the shifting between the two signal groups has to be taken into account. Notice that for the intra-node offsets, too, the traversal direction has to be minded. For a more evolved example, see Fig There, all relevant time gaps are depicted through arrows, i.e., directed arcs. Consider now, for example the point in time when the green phase of the first signal group of Signal 1 begins. Starting from that point, following the link offset arcs and the intra-node arcs one can traverse a cycle. Thus, for this example, the sum Ψ φ 13 Ψ Ψ 1 3 φ 23 + Ψ φ 24 Ψ Ψ 3 4 φ 14 Ψ 3 1 (1.7) must equal an integral multiple of the cycle length, since otherwise no consistent setting of the node offset at Signal 1 is possible. So, Equation (1.6) in fact turns out to amount to a F(l) φ a a R(l) φ a + a F(l) ( Ψ os(a) u ) Ψ ds(a) v ( Ψ os(a) u a R(l) ) Ψ ds(a) v = n l c, (1.8)

32 PSfrag replacemen 26 The Network Signal Coordination Problem Ψ 2 1 Ψ 3 1 φ 13 φ Ψ 2 2 Ψ Figure 1.8: An example network G = ({1, 2, 3, 4}, {(1, 3), (2, 3), (2, 4), (1, 4)}). In bluish colors, the link offsets are sketched whereas the intra-node offsets are depicted in red. Ψ 1 3 φ 14 φ Ψ 2 4 Ψ where os(a) and ds(a) again denote the signal groups at the origin and destination of the link a = (u, v). An important observation is that, although the cycle equations (1.8) have to hold for each cycle l L, they do not have to be defined explicitly for all cycles. Already in 1966, Little [Lit66] observed that it suffices to define the cycle equations for the cycles that are induced by the chords of a spanning tree. Later, in 2002 Liebchen and Peeters [LP02] introduce the class of integral cycle basis. Moreover, they show that it suffices to require Equation (1.8) for the elements of an integral cycle basis [LP02] in order to ensure that it holds for all cycles of the graph. Note that the class of integral cycle bases is a proper superset of the class of strictly fundamental cycle bases that are defined by spanning trees [LR07]. A second important observation related to the cycle equations (1.8) concerns the integer variable n l, for l L. For computational issues it is important to provide bounds for them. However, with the help of the bounds on the link offsets, (1.5), one can easily define an upper bound n l and a lower bound n l on n l. Namely, we get n l = 1 c (r a + τ a ) (r a + τ a c) + Ψ (1.9) a F(l) a R(l) and n l = 1 c (r a + τ a c) (r a + τ a ) + Ψ a F(l) a R(l) (1.10) as trivial bounds. Note that the terms Ψ are a shortcut form the extended form as in (1.8). With these bounds on the integer variables at hand, one can consider

33 1.3 A Revised MIP Formulation for the NSC problem 27 the product that was also denoted width of a cycle basis in [Lie06], ( nl n l + 1 ) (1.11) l B as quality measure of a basis B for the computation of the MIP. Hence, as we can choose any integral cycle basis for the MIP formulation, it seems promising to take the smallest one with respect to (1.11). However, we are only aware of one single contribution, see Liebchen [Lie06, LPW05], that ever tested the actual influence of a cycle basis on the computation time of the MIP. In Sect. 4.1 we provide a second computational study that illuminates the correlation between the quality of a cycle basis and the computational performance of the according mixed-integer linear program A Variable Phase Sequencing In contrast to the approach by Gartner et al. [GLG76], our model offers the possibility to choose between different red-green split modes, i.e., we allow a variable phase sequencing. In the previous sections we assumed a fixed signal timing plan at the signals for clarity reasons. In this section we formally introduce the variable phase sequencing. This, however, may not be confused with an entirely free choice of lengths of red and green phases. In detail, the variable phase sequencing works as follows. For a vertex v V, R v denotes the set of signal groups at v. Then, we consider α v different predetermined modes of operation at the signal v,i.e., α v different signal timing plans. Nevertheless, these different signal timing plans must have some properties. First, the signal groups have to stay the same, i.e., throughout all modes, the turns at a junction that are controlled by a signalgroup p R v stay the same. With this, the possibility of an assignment of the traffic to signalgroups at the upstream and at the downstream signal is kept. A second property is that the lengths of the green phases and red phases are the same for all modes. Thus, the variable phase sequencing offers the functionality of choosing between different modes each giving different intra-node offsets to the signalgroups. In particular, the parameter Ψ v,p m denotes the intra-node offset at junction v of signal group p R v in mode m, where m = 1,...,α v. Then, we introduce binary variables d v,m for all nodes v and modes m = 1,...,α v. These variables manage the selection of a mode. Namely, with Equations (1.12), the actually chosen intra-node offset of signalgroup p at node v is allocated to the variable Ψ v,p : α v m=1 d v,m Ψ v,p m = Ψ v,p v V, p R v. (1.12) Of course, one has to ensure that exactly one mode is adjusted. This is done in Equations (1.13). α v m=1 d v,m = 1 v V. (1.13)

34 28 The Network Signal Coordination Problem With Figure 1.9 we give an impression of how such different modes or the according signal timing plans could look like. Ψ 2,3 1 Ψ 2, v v v v Ψ 2,4 3 Figure 1.9: An example of different predetermined signal timing plans, i.e., modes, is shown. Exemplarily, some of the Ψ are indicated. Ψ 2,1 4 Observe that the introduction of variable timing plans, and thereby changing the status of the intra-node offsets from parameters, as introduced before, to variables, is well compatible with the cycle equations proposed in Section In particular, by the according variables Ψ u,p and Ψ v,p. Note that here it is important to still be able to assign an edge to its signal group. However, when considering the definitions (1.9) and (1.10) of the bounds for the integer variables for the cycles, we observe that these bounds become much weaker. in (1.8) one only has to replace the parameter Ψ os(a) u and Ψ ds(a) v The Objective Function In this section we introduce the objective function. Our mathematical model is built to minimize the total traffic network delay that is due to missing or poor coordination within the network. Now, we will specify how the delay is actually determined by evaluating the arrival pattern of the platoons. First, remember that due to the assumption of the vehicles moving in platoons through the network, we have a separable objective function. Thus, the delay on a link does only depend on the offset of that link. Therefore, consider the link a = (u, v) A and assume certain signalgroups at the origin of the link and at the destination. Now, the traffic, i.e., the platoon, leaves the junction u. Namely, throughout the green phase of signalgroup os(a) at the origin signal u, the platoon is released. Then, the platoon is moving on the link towards signal v. The transit time on that link is determined by the head, i.e., the first vehicle, of the platoon. See Figure 1.10(a) for an illustration. Finally, the platoon arrives at signal v. Observe

35 1.3 A Revised MIP Formulation for the NSC problem 29 that the length of the platoon then may have changed. This means that although the platoon was loaded for a timespan with length equal to the length of its green phase at u, the length of the platoon can have changed while traversing the link. This is due to dispersion effects on the link that appear especially on links that represent long streets. However, note that the actual necessary information for calculating the τ a v 2 γ a v (a) u Figure 1.10: In Figure (a) we schematically depict the platoon moving from Signalgroup 2 at Signal u to Signalgroup 2 at Signal v. In (b) we focussed on the detail that is considered when evaluating this link s delay. Notice that all information about intra-node offsets are left out. Only the arrival time, γ, of the platoon has to be known. delay are the arrival time γ a and the length of the platoon at its arrival at signal v. The effects that change the platoon shape in between are not pursued. Further, for the calculation of the delay on link a, we slightly change the perspective, see Figure 1.10(b). Namely, we consider the timespan of length c not as it is given from the signal timing plan, but beginning with the red phase and followed by the green phase, as indicated in Figure 1.10(b). This changed viewpoint is also recognizable in Fig Then, the situation is as follows. The platoon can pass the signal as long as it shows green. If the signal shows red, the vehicles have to stop and a waiting queue is formed. When the signal turns green again, the queue is released. For an illustration of this queueing see Figure 1.11(a). Now, as we want to quantify the delay, we have to introduce some parameters. As an input of the model, we are given in advance the traffic volume on the links. So, with f a indicating the number of vehicles on link a = (u, v) A, which we measure in vehicles per hour, it is straightforward to calculate a vehicle arrival rate fp a within the platoon. Namely, it is defined as fp a = (b) c f a 3600 p a. (1.14) Notice that we assume a uniform rate with the platoon. So, this rate fp a is the rate with which the waiting queue is built up, when there is a red at signal v. Then, when the signal turns green, the queue is released with a saturation rate of s a. However, if at the beginning of the green phase there are still vehicles arriving, see the situation that is depicted in Figure 1.11(b), the net-saturation rate is s a fp a.

36 30 The Network Signal Coordination Problem A 1 A 2 A 3 A 4 (a) (b) Figure 1.11: Depending on the arrival time of the platoon, the vehicles queue at the signal. For example, if the platoon arrives at the destination signal at the beginning of the green phase, (c), and additionally the platoon is not longer than the green phase, then no delay arises. (c) Applying these straightforward ideas of how the waiting queue behaves, makes a calculation of the occurring delay easy. When interpreting this process geometrically, determining the delay is nothing else but calculating the size of the area that visualizes the queue over time in function of the arrival of the platoon relative to start of red phase. See for an example Figure 1.11(a) where the arrival time of the platoon is γ a = 0. We denote with Z( ) (z( )) the function that measures the total (average) delay of that link in dependence of the arrival time. Then, we observe the following total delay for that particular example, 7 Z(0) = A 1 + A 2 + A 3 = p2 a fp a 2 + (r a p a ) p a fp a + p2 a fp 2 a 2s a. (1.15) Then, of course, this quantity has to be normalized with the factor p a fp a in order to obtain the delay per vehicle on link a with γ a = 0. At this point, we abstain from providing all possible cases, i.e., explain how Z( ) is calculated for other values of γ and with different assumptions on how, e.g., p a is related to r a and g a. Nevertheless, an important assumption is the following: we require that the waiting queue must have dissipated completely when the signal turns red. Without such an assumption, considering a time span of length c would not be sufficient to determine the delay. Now, how do we use the function z( ) to obtain an objective for our mixed-integer program? A first observation is that we cannot use the above described function z( ) as objective function, because it is not linear in the arrival time γ. In fact, the function z( ) is piece-quadratic in γ which can be seen from Figure 1.11(b). In such a situation, i.e., the arrival time γ a lies for example in the interval [r a p a 2, r a + p a 2 ], the size of the area depicted with A 4 is quadratic in γ a. As a consequence, we will use a piece-wise linear approximation of z( ) as follows. We evaluate in advance the function z( ) for particular γ i and use the points (γ i, z(γ i )) for a linear approximation. Note first that z(0) = z(c). Further, it is obvious that for γ = 0 the periodic function z( ) attains its maximum. On the other hand, for γ = r a, z( ) is minimal. Thus, at the best, the platoon arrives exactly when the signal turns green. 7 assuming p a r a

37 1.3 A Revised MIP Formulation for the NSC problem 31 Notice that if p a > g a we do not have a delay of zero even when γ a = r a. However, this is not a contradiction to the assumption that the queue must have dissipated when the signal turns red, e.g. when the net-saturation rate s a r a is high. delay 0 r a c (a) 0 r a c (b) Figure 1.12: In (a) a link performance function is depicted. The technique to get to that link performance function is illustrated in (b). So, for each link we use between 3 and 5 data pairs to define the objective. We always consider the pairs (0, z(0)), (r a, z(r a )) and (c, z(c)). Additionally, we add up two more pairs with γ-value slightly smaller or slightly larger than r a. Then, every two consecutive data points are used to define a line. See Figure 1.12 for an example. That means, for a link a = (u, v), let g a k ( ) denote the lines, with k = 1,..., β a, where β a stands for the number of lines defined for link a. Now, we introduce the variable z a for a link a A for the approximated average delay on link a. Then, we add the inequalities z a g a k (γ a) k = 1,...,β a, (1.16) for all a A to the constraint set of the MIP. Thus, we force the average delay z a to be at least as big as the approximated delay for all sections, which is defined by the particular line. Of course, if and only if the piece-wise linear function is convex, Equations (1.16) enforce that z a takes a value that really corresponds to an approximated delay. Notice that this condition on the piece-wise linear delay function to be convex is the reason why we switched the perspective, as seen in Fig. 1.10(b). Herewith, we shifted the maxima of the delay function z( ) to the lower and to the upper bound on the values, the variable γ can take. Finally, the objective function of our MIP is defined by f a z a. (1.17) a A We conclude this section with a remark: it is unproblematic to introduce non-uniform traffic rates for the platoon, which, however, has to be done in advance. Alone, while preprocessing, the calculation of the delay Z( ) becomes more evolved, since the geometric visualization of the waiting queue has a more complicated shape and one has to distinguish more cases.

38 32 The Network Signal Coordination Problem Non-uniform Cycle-lengths In this section we discuss how we relax the assumption of a network wide uniform cycle length at the signals. Especially when considering medium or large size networks, claiming a uniform cycle length for the signals of the network may appear restrictive. In a network, not all junctions are similar with respect to their layout or the amount of different traffic streams their signal has to cope with. Often, there are signals with a large number of signal groups they control many competing traffic streams which therefore need a large cycle length. On the other hand, for a small junction with, say, just two competing streams, a short cycle length is possible and, thus, preferable, since less delay is the direct consequence. So, a first idea is to introduce variables for the cycle lengths that adjust optimal cycle lengths at the signals or at least one global variable cycle length for all signals. However, incorporating variable cycle lengths or even just one variable cycle length into the approach presented so far, is problematic. A major drawback in, for example, an extension by Gartner et al. [GLG75b] of their approach to the network coordination [GLG75a], is the definition of the objective, also see Section With a uniform cycle length in the network, it was sufficient to approximate the actual delay with lines. Including a variable global cycle length into the model, however, would change the delay function Z( ) to a function Z (, ) of two unknowns: the arrival time and the cycle length. Thus, one would have to approximate Z (, ) by planes. Although in [GLG75b], the authors present an approach to do so, we do not see how the convexity of the approximation of Z (, ) can be ensured for all relevant cases. We propose a different possibility to relax the condition on a network wide prescribed uniform cycle length. We allow for arbitrary cycle lengths at the signals, which, though, have to be given in advance, i.e, they are input parameter and not variables. Formally, we denote by c v the cycle length of signal v V. In the remainder of this section, we elaborate on how these non-uniform cycle lengths are incorporated into the model. A first and general observation is the following. The quantity lcm({c v v V }), i.e., the least common multiple of the cycle lengths, can be seen as a global cycle length for the signalization in the network. That means, after that amount of time, the signalization pattern of the entire network repeats. However, this new concept of periodicity can be investigated better when considering a single edge. So, consider an edge a = (u, v) A. The signal u operates with a cycle length of c u, whereas signal v has cycle length c v. Hence, after a timespan of length lcm(c u, c v ), the signalization pattern for link a repeats. Formally, this is nothing else but the following observation. For link a = (u, v), the node offsets π u and π v induce the same situation, i.e., result in the same delay for the vehicles on that link, as the offsets π u and π v, with π u := π u + p a gcd(c u, c v ) (1.18) and p a being an arbitrary integer. Further, with defining the cycle length for a

39 1.3 A Revised MIP Formulation for the NSC problem 33 link a = (u, v) A as c a := gcd(c u, c v ), the connection between the node offsets π u and π v and the link offset φ a is π u + Ψ os(a) u + φ a Ψ ds(a) v + p a c a = π v. (1.19) In contrast to the case of uniform cycle lengths, see Equation (1.2), we now face a problem. Having at hand a solution consisting of link offsets φ, the node offsets π are not uniquely determined via Equation (1.19). For example in [Hae04] it was shown that when propagating the link offsets φ, e.g., along a spanning tree, and thereby fixing the π simply to any possible value, may lead to a contradiction. However, we report in Section on how one can calculate the node offsets with the link offsets; also see [Nac96b]. The next question to discuss is, how the offsets can be coupled to the arrival time γ of the platoon. Let a = (u, v) A. Then, lcm(c u, c v )/c u many different arrival times at signal v in a period of length lcm(c u, c v ) can be observed. The timespan between these different arrival points is equal to the cycle length c u at node u. See Figure 1.13 for an illustration. However, it suffices to consider exactly one arrival time γ, which then again is denoted by γ a. In particular, we model the π v γ γ c u φ π u τ Figure 1.13: The connection between the offsets and the arrival time γ for nonuniform cycle length case is depicted. In this example we consider an edge a = (u, v) A. The cycle lengths fulfill 3 c v = 2 c u. connection between γ a and the link offset φ a as τ a γ a + r a = φ a, a A. (1.20) Note that this is exactly the same constraint as in the uniform cycle length case, (1.4). However, we additionally have to give bounds in order to make the relation between the arrival time γ a and the offset φ a uniquely interpretable. Namely, we set γ a [0, c a ] and φ a [τ a + r a c a, τ a + r a ]. Notice that this way we allow for exactly one value for the φ a variable and one value for the γ a variable. Moreover, consider the physical constraints that are modeled by the cycle equations. See Equations (1.8) in Section For a cycle l L, define now the cycle

40 34 The Network Signal Coordination Problem length of l, denoted by c l as the greatest common divisor of the cycle lengths at the signals v V that are contained in l. Namely, set c l := gcd({c v v l}). (1.21) Then, the cycle equations for the uniform cycle length case, (1.8), canonically translate to the non-uniform cycle length case. For an l L, we demand a F(l) φ a a R(l) φ a + a F(l) ( Ψ os(a) u ) Ψ ds(a) v ( Ψ os(a) u a R(l) ) Ψ ds(a) v = n l c l. (1.22) Again, similar to Lemma 1.4, the following can be observed for a link offset vector φ É A : if there exists a vector π É A for which Equations (1.19) hold, then using the fact that c a /c l is integral also Equations (1.22) hold for all l L. On the other hand, if (1.22) holds, one can find a vector π É A for which Equations (1.19) hold [Hae04]. Moreover, note that the previous fixing of the link offset φ a to the interval [τ a + r a c a, τ a + r a ] did not constitute a restriction; again because c a /c l is integral. In the case of uniform cycle lengths it was sufficient to formulate the Cycle Equations (1.22) for the elements of an arbitrary integral cycle basis, e.g., for the elements of a strictly fundamental cycle basis induced by an arbitrary spanning tree. However, when the signals operate with different cycle lengths, not any integral cycle basis can be used, see [Hae04]. In particular, choosing an arbitrary spanning tree does not work out in general. On the other hand, Haenelt [Hae04] observes that there always can be constructed an integral cycle basis B such that if (1.22) holds for all l B, then it holds for any cycle of the graph. A next observation is that the non-uniform cycle lengths at the signals do not affect the modeling of variable phase sequences: the Constraints (1.12) and (1.13) can be adopted. Finally, let us consider the objective function in the non-uniform cycle lengths case. With a = (u, v) A, we have to consider a timespan of length lcm(c u, c v ) in order to determine the delay on link a. See Figure 1.14 for an illustration. Again, Figure 1.14: A possible arrival pattern of a link (u, v) A with lcm(c u, c v ) = 4 c u = 3 c v is shown. For example, think of c u = 60 seconds and c v = 80 seconds. the delay is determined by calculating the size of the area that represents the waiting queue. The actual computation works just the same as in the uniform cycle length case, see Section We again evaluate delay functions z( ) and Z( ), respectively, at prescribed arrival times γ i and use the data pairs to build up a piece-wise linear

41 1.3 A Revised MIP Formulation for the NSC problem 35 continuous link performance function. Note that instead of (1.14), we use fp a = c u f a 3600 p a (1.23) to determine the traffic rate in vehicles per second within the platoons. Then, the constraints (1.16) and the objective function (1.17) can be adopted from the uniform cycle length case. Finally, in Table 1.2 we show the particular platoon lengths that we used to calculate the functions z( ) and Z( ), see (1.15). These values assume links with two lanes and are to be adapted when the network layout is different. Table 1.2: The table shows the grading of the platoon lengths in function of the traffic volume. One single link is considered and the traffic volume is given in vehicles per hour. Platoon lengths p in dependence of traffic volume f Traffic volume p in s Traffic volume p in s 0 f < f < f < f < f < f 23 We conclude this section giving some remarks. A first observation is that adding the functionality of non-uniform cycle lengths to the model, does not increase the model in terms of number of constraints and number of variables. The preprocessing step to define the linear functions gk a ( ) for the objective becomes more involved, which is, though, computational negligible. Furthermore, it has to be mentioned that in the non-uniform cycle lengths case, it is more difficult to provide piece-wise linear functions that are convex. Remember that the convexity for the link performance functions is essential for the model. The reason for this difficulty is the following. In the uniform cycle lengths case it was obvious that for a link a A the link performance function takes its maxima at the according γ a s lower and upper bound, 0 and c, respectively. Further, a minimum was attained at γ a = r a. In case of non-uniform cycle lengths in the network, maxima and minima of the link performance functions cannot be located immediately. Although, theoretically, there is no reason to restrict the values for the cycle lengths, one should keep the quantity lcm({c v v V }) gcd({c v v V }) (1.24) small. Otherwise, e.g., if there are u, v V with gcd(c u, c v ) = 1, the coordination of the signals becomes degenerated since the instance has a smaller degree of freedom 8. For example, if there is a link a between u and v, any value for the offset φ a brings the same delay for link a. 8 Assume for this example that we consider integer cycle lengths and offsets.

42 36 The Network Signal Coordination Problem However, we suggest a precondition on the cycle lengths. Namely, allowing three values as cycle length, for example c v {60, 80, 120}, should be a good compromise between a realistic mapping of requirements of real world traffic networks and computational issues. Note that allowing (fixed) non-uniform cycle lengths within an approach for coordinating signals in a network can be considered novel. In [GLG75b], Gartner et al. presented a quadratic approach for coordinating signals with a variable but uniform cycle length. Further, Serafini and Ukovich [SU89b] also gave a problem formulation without an objective function that included a variable but uniform cycle length at the signals. Moreover, Serafini and Ukovich [SU89a] introduced an extension to their Periodic Event Scheduling Problem (PESP), see Section 1.2.2: the so-called extended PESP (EPESP) where non-uniform but fixed cycle lengths are allowed. For example Nachtigall [Nac96b] and Haenelt [Hae04] discuss modelling issues of the EPESP The Mixed-Integer Linear Program In this section we finally state the mixed-integer linear program for the NSC problem in which we optimize the coordination of fixed-time traffic signals. The MIP contains the variables and constraints that we developed in the Sections to In the remainder of this thesis we refer to this MIP as NSC MIP or simply by (1.25): a F(l) φ a a R(l) min a Af a z a (1.25a) α v m=1 φ a + Ψ = n l c l l B, (1.25b) d v,m Ψ v,p m = Ψ v,p v V, p R v, (1.25c) α v m=1 d v,m = 1 v V, (1.25d) z a g a k (γ a) a A, k = 1,...,β a, (1.25e) τ a γ a + r a = φ a a A, (1.25f) n l n l n l l B, (1.25g) n l, γ a [0, c a ], d v,m {0, 1}. Here, Ψ in (1.25b) is a shortcut for ( Ψ u,os(a) Ψ v,ds(a)) a=(u,v) F(l) a=(u,v) R(l) ( Ψ u,os(a) Ψ v,ds(a)). (1.25h) To summarize: Equations (1.25b) are the cycle equations on the offsets. Remember that B is an integral cycle basis: an arbitrary one in the case of uniform cycle lengths at the signals and a specifically constructed one, see [Hae04], otherwise.

43 1.4 Application of the NSC Model in Practice 37 Further, Equations (1.25c) and (1.25d) manage the selection of a mode for each signal, i.e., the selection of one particular prescribed signal timing plan. Moreover, Equations (1.25f) couple the offsets φ with the arrival times γ, which are then via Inequalities (1.25e) used to determine the average delay z for a link. Finally, (1.25g) and (1.25h) formulate bounds, cf. (1.9) and (1.10), and the integrality of the cycle variables. However, after having solved (1.25), an important task remains. A solution of (1.25) contains, among others, a set of offsets φ a and arrival times γ a for each a A. Nevertheless, as we want to evaluate the MIP, we are interested in node offsets. Hence, one has to carry out a backtransformation of the link offsets φ a to node offsets π v. For describing how such a backtransformation works we distinguish two cases. First, assume that there is a uniform cycle length at the signals, i.e., c u = c v for any u, v V. Then, backtransformation is done simply by propagating the link offsets along the edges of a spanning tree T of G where one node offset was fixed before. This is done using Equation (1.2). The key observation to see that this works is that the cycle constraints (1.25b) ensure that (1.2) holds for the non-tree edges of T. Of course, for that propagation, intra-node offsets have to be considered. On the other hand, when facing different cycle lengths at the network s signals, backtransformation is more complicated. In general, propagating the link offsets along a spanning tree is not possible anymore, because with (1.19), the node offsets are not uniquely determined by the link offsets φ a. Instead, one has to solve the mixed diophantine linear equation system consisting of (1.19) for all a = (u, v) A. This can be done algebraically or by solving a separated MIP. To the best of our knowledge, there is no efficient combinatorial approach for the backtransformation of the link offsets to node offsets. In Section 1.5 we discuss possible enhancements of the model. Also notice that in Chapter 4 we conduct experiments related to the NSC MIP (1.25). In particular, we will evaluate the MIP at some real-world example networks. Moreover, we test the correlation between cycle bases used for the MIP formulation and the MIP performance. In addition, we use the real-world instances to conduct a MIP solver comparison. 1.4 Application of the NSC Model in Practice In this section we address the issues of an NSC model in general and the NSC MIP (1.25) in particular. The popularity of signal timing optimizers, such as TRANSYT [Rob69] and SYN- CHRO [syn00], lends practical relevance to the network coordination problem. Such optimizers accept as input a description of the supply, i.e. the street network in the study area (mainly links with travel times, permitted maneuvers at junctions), and of the demand, i.e. traffic flows through the network. The demand can be either observed or derived from a demand model. Interestingly, demand models are themselves extending in the direction of traffic engineering, supporting models for signal

44 38 The Network Signal Coordination Problem control at junctions and offering analysis methods for them, e.g. by incorporating capacity analysis according to the HCM [hcm00]. These models aim to provide a more realistic node impedance for route choice by incorporating capacity analysis into the assignment step. So far, most packages are limited to the optimization of cycle lengths c and green time fractions (g/c), as run time requirements have precluded network coordination. This is due to the nature of the solution methods often used in practice (genetic algorithms or quasi-exhaustive search). A faster solution method would enable an even closer integration between macroscopic demand modeling and network coordination. The Network coordination could then be part of the assignment process, allowing to update route flows in response to changed offsets, and thus reach an equilibrium between demand and supply, including all aspects of signal control. Figure 1.15: The organization of the overall optimization process with components assignment, local optimization and network coordination. We focus on the latter one stressing the requirements in the context of the optimization scheme. This overall optimization process is illustrated by Fig Given OD flows (from previous model steps) are assigned to a network with signal control, using an assignment method which calculates node impedance from signal timings and offsets between different junctions. The assignment produces node flows (traffic volumes per maneuver at each junction) which are input to the local optimization of g/c. Path-based assignment procedures also produce route flows on the subpaths between successive signalized junctions, or more specifically, between successive signal groups or stages. In addition the assignment yields the volumes of non-platoon traffic arriving at a signalized junction from non-priority un-signalized maneuvers upstream. A network coordination optimizer takes these sub-path flows and globally optimizes the offsets. Then, the set of offsets are fed back to assignment. The loop is executed at least once and continues until route impedances (including loss times experienced at signal controls) and flows converge. However, before the application of the network coordination presented in this

45 1.5 Conclusion and Open Questions 39 section can actually be realized in practice, some open issues have to be discussed. A first question is how the coordination of the signals is actually distinguishable within the assignment step, see Figure Therefore, one has to develop traffic assignment models that include offsets of signals into their impedance functions. A second important issue is the one of the convergence of this iterative procedure illustrated in Fig It must be assumed that any strict claim on convergence behavior will not be easy to prove. However, let us remark that an application of network signal coordination in general and the MIP (1.25) in particular within the iterative procedure, Figure 1.15, is not the only possible application. Rather, the optimization of the coordination of signals in a network is an important task in itself, i.e., the usage as a stand alone tool is a relevant application, too. Depending on the respective application, different requirements on the model s computational tractability arise. Whereas for the application of a network signal coordination approach computation times in the timescale of seconds are required, practitioners consider computation times of several hours for a stand alone application to be reasonable. 1.5 Conclusion and Open Questions In this section we considered the Network Signal Coordination (NSC) problem. After a short introduction of the most important terms related to traffic signals, we briefly reviewed previous work on coordinating traffic signals. Then, in Section we gave a formal mathematical problem definition of the NSC problem. For this definition we ensured that certain phenomena of the problem in practice are covered. Thereafter, we investigated connections between the NSC problem and related problems such as the Periodic Event Scheduling Problem and stated an NP-completeness proof for the NSC problem. Then, during Section 1.3, we derived in detail a revised MIP formulation for the NSC problem. Finally, we reported on possible applications for a network signal coordination model. In the remainder, we discuss open questions and tasks related to the NSC problem in general, and our MIP approach in particular. We decided to build a model for the c-nsc that uses offset variables on the edges instead of node offset variables. The reason for doing so was the objective function. Nevertheless, it would be interesting, whether a MIP approach that together with the link offsets involves node offsets, performs comparable. Experiences from the PESP context, show that MIP approaches with variables for the nodes sometimes computationally behave well. When considering our NSC MIP (1.25) several possible extensions come to mind. For example, variable cycle lengths, variable lengths of red phases and green phases and even variable travel speeds for the vehicles are topics for further research. However, incorporating these additional functionalities, would cause major changes of the model and is not at all trivial to accomplish.

46 40 The Network Signal Coordination Problem Moreover, an important question when building up a model for network coordination, is, first, the treatment of signals that are not predestined for the optimization of offsets, e.g., traffic responsive signals, and second, the treatment of unsignalized junctions. This remains an issue, since in almost all inner-city traffic networks of relevant size, one can find unsignalized junctions and traffic responsive signals together with fixed-time signals. In this very same context, the treatment of public transport means, which often actuate signal controllers, is to be mentioned as an open task. However, when considering the possible application of network coordination of Section 1.4, one observes a limitation of the network coordination approach. A question that arises is why should one separate the network coordination from the traffic assignment?. A merging of these two optimization tasks can be motivated from both sides. First, from the viewpoint of network coordination optimization, it is not entirely realistic to assume that the vehicles move on prescribed paths through the network. Rather, it can be assumed that for example a coordination that penalizes some paths, actually changes the drivers route choice. Second, from the viewpoint of traffic assignment, it is unrealistic not to consider signal settings when defining node impedance functions. And, of course, the coordination of signals indeed is an important signal setting. Although the idea of an integrated model of traffic assignment and signal coordination is, of course, not new, so far, only approaches merging the traffic assignment with models concerning traffic signal settings different from coordination, were presented. Namely, a main difficulty seems to lie in incorporating non-local quantities like the coordination into an traffic assignment approach. Thus, since a fully integrated model seems rather complicated at first sight, one could think of the following. A first step could be including the offsets as parameters into an traffic assignment model, which is necessary to process the iterative optimization scheme (Figure 1.15). That means investigating and quantifying the influence of a coordination of traffic signals to travel times and route choice of drivers in networks.

47 2 STRICTLY FUNDAMENTAL CYCLE BASES ON GRIDS In this chapter we are going to investigate strictly fundamental cycle bases on square planar grid graphs. We develop both new lower and upper bounds for the problem. Moreover, we conduct some computational experiments that improve on the experimental studies carried out so far. This chapter is based on [KLRW08] and [LWK + 07]. 2.1 Introduction Suppose T is a spanning tree in a graph G; the fundamental circuits with respect to T form a strictly fundamental cycle basis (see Section 2.2 for formal definitions). We refer to the problem of finding a spanning tree, in which the lengths of its fundamental circuits sum to a minimum value, as the Minimum Strictly Fundamental Cycle Basis (MSFCB) Problem. As a generalization, in the Minimum Cycle Basis (MCB) Problem one seeks a general cycle basis of minimum length. Applications. There are a variety of applications for the MCB problem. These include biology and chemistry [Gle01], traffic light planning [KMW05], periodic railway timetabling [Lie06], and electrical engineering [Bol02]. Typically, cycle bases are computed during a preprocessing phase. During the actual computations one ensures that a certain problem-specific property is true for the elements of the selected cycle basis in the graph of interest. By the properties of cycle bases, one can conclude that this particular property is actually true for any circuit in the graph, just as it is required by the practical application. In many cases it can be observed that shorter cycle bases imply a shorter time for the actual computations. For some of these applications not all cycle bases are of use (e.g., traffic light scheduling and periodic railway timetabling); however, strictly fundamental cycle 41

48 42 Strictly Fundamental Cycle Bases on Grids bases being the most specialized ones always are. In other applications, such as electrical engineering, it is generally much more favorable to use strictly fundamental cycle bases, because of the numerical stability of the subsequent calculations [BE05]. The practical relevance of the MSFCB problem is reflected by numerous computational studies by different groups working in combinatorial optimization [ALMM04, DKP82, DKP95, GC67, LAM05, Pat69]. Theory. Already in 1982 Deo et al. [DKP82] proved the MSFCB problem to be NP-hard for general unweighted graphs. Their proof can easily be extended to bipartite graphs, see [Lie06]. Because of the practical relevance of these cycle bases for various applications, many heuristics were proposed and tested. However, for none of these heuristics, any non-trivial approximation ratio or any non-trivial bound on the absolute length of the resulting bases was shown. Very recently, Galbiati et al. [GRA08] showed that the MSFCB problem is APX-hard, even when restricted to unweighted graphs. With respect to the length of an MSFCB, Deo et al. [DKP82] conjectured that any unweighted graph on n vertices has an MSFCB of length O(n 2 ). Then, Abraham et al. [ABN07] came up with a construction that can be used to derive a positive answer to this conjecture [ELR07]. For non-dense graphs a result by Elkin et al. [EEST05] provides an even better bound: O(m log 2 nlog log n), where m denotes the number of edges. Notice that they originally considered the averagestretch tree spanner problem. However, profiting from the Unified Notation for Tree Spanner problems (UNTS, see [LW07] or Chapter 3) one can conclude that in the case of unweighted graphs their results can be applied immediately to the MSFCB problem. For complete graphs with arbitrary weights and for metric graphs, Galbiati et al. [GRA08] provide polynomial-time approximation schemes. For N N grid graphs Alon et al. [AKPW95] showed that optimal trees for the MSFCB problem induce bases of length Θ(nlog n), with n = N 2 being the number of vertices of the grid. Why Planar Grids? Due to the absence of theoretical bounds for many heuristics, the authors of these approaches used empirical calculations to evaluate the quality of their algorithms. Yet, to compare different heuristics empirically, it is essential to run them on the very same input graphs. But what are good testbeds? Liberti et al. [LAM05] consider square planar grid graphs being the most difficult testbeds for the MSFCB problem, both for heuristic and exact methods, due to the huge quantity of configurations having the same strictly fundamental cycle basis (SFCB) cost. For example, using our result in Section one can show that there exist more than 80,000 optimal solutions for the MSFCB problem on the 8 8 grid compare this to the input size of only 112 edges! In fact, also from a theoretical point of view difficulties can be motivated in several ways. First, planar grids are almost regular (more than n 4 n vertices have degree four) and within a fixed distance, the subgraphs around almost each vertex are isomorphic. Hence, any heuristic that bases important decisions on local

49 2.1 Introduction 43 configurations is likely to perform poorly. Second, if G was a tree, then in the MSFCB problem no decisions are to be made and the problem clearly becomes trivial. An appropriate measure for the tree-alikeness of a graph is its treewidth [Bod93]. With respect to that measure, grid graphs having Θ(n) edges and treewidth n are prominent examples of being far away from being a tree [RS86]. Thus the MSFCB problem is likely to maintain its combinatorial hardness when considered on square planar grids. Finally, also from a practical perspective planar grids are very suitable. Many of the relevant instances in several applications are planar graphs or even planar grids (e.g., electrical engineering and traffic light scheduling). Focusing on grid graphs could appear narrow. But it is commonly believed that these hold the key to better algorithms for cycle bases. Indeed, for square planar grid graphs Alon, Karp, Peleg, and West [AKPW95] designed spanning trees that induce cycle bases of length 4 3 nlog 2 n + O(n), see Section 2.4. They proved that these trees are asymptotically optimal. Using this asymptotic upper bound we demonstrate how heuristics that base major decisions on local configurations risk failure. For instance, the degree-based C-order heuristic [LAM05] can be implemented to compute so-called Machete-trees (cf. Section 2.4 and Figure 2.10 for example trees). Although these trees minimize the maximum stretch, they yield a poor MSFCB objective value of only Θ(n 3 2). In particular, such heuristics may miss the optimum by a non-constant factor. This again underlines the property of grids being a relevant testbed. In fact, Liberti et al. [LAM05] also select grid graphs as one of their testbeds. On the grid they observe that their new C-order heuristic attains an objective value of 46,452, showing that the value 48,254 of the NT heuristic [DKP95] is even worse. Unfortunately, such an isolated comparison does not clarify whether these are indeed good absolute objective values. Amaldi et al. [ALMM04] consider grid graphs in their computations as well. They report a solution of objective value 23,026 for the same grid-size, and it was obtained by local search techniques. Liebchen et al. [LWK + 07] further improved this value to 21,920 and is it not clear which further improvements are possible. In other words, there is a need for good benchmark values for the MSFCB problem for the particularly challenging case of planar grid graphs also for the future evaluation of new heuristics. Contribution. Alon et al. [AKPW95] established that an optimal solution for the MSFCB problem on grid graphs is of length Θ(nlog n), with n beeing the number of vertices in the graph. In detail, they proved ln 2 2,048 nlog 2 n O(n) to be a valid lower bound 1 and proposed trees that achieve 4 3 nlog 2 n + O(n). In this chapter we improve on both of these bounds. Namely, in Section we introduce a lower bound of 1 12 nlog 2 n O(n) whereas in Section we provide a family of spanning trees with length no more than nlog 2 n + O(n). However, when we consider small grids this new asymptotical lower bound is not applicable. A lower bound for the MSFCB problem that is not restricted to planar grids is the length of a general minimum cycle basis, and it was used for example by 1 Alon et al. were not trying to optimize the constants.

50 44 Strictly Fundamental Cycle Bases on Grids Amaldi et al. [ALMM04]. Applying this MCB bound to square planar grids yields a lower bound of 4n 8 n + 4. In Section we exploit the particular structure of grid graphs to come up with a new lower bound of 6.25n 23.5 n + 34, which constitutes the best known lower bound for small grids. However, this bound is not tight for 8 8 grids. Hence, in Section we present a combinatorial proof for a tight lower bound for the 8 8 grid. Furthermore, in Section we develop additional valid inequalities to a mixedinteger linear programming approach for the MSFCB for general graphs by Amaldi et al. [ALMM04]. In addition, we present a new MIP model in Section that is tailored for square planar grids. Then, in Section 2.5 we conduct experiments and provide benchmarks for upper and lower bounds for the MSFCB problem on grids up to the size of These results improve on previous ones by [ALMM04] and [LAM05]. Finally, we present a gallery of best known spanning trees for sizes 9 9 up to Prelimenaries We consider cycle bases of a 2-connected simple undirected graph G = (V, E). Define n = V, m = E, and ν = m n + 1, where ν is the cyclomatic number of G. Let C be a circuit (cf. [Sch03, Ch. 3]) in G and denote by γ C its {0, 1}-incidence vector. The cycle space C of G is the following vector subspace over GF(2): C := span({γ C C circuit in G}). A cycle basis B of G is a set of ν circuits of G whose incidence vectors are a basis of C. The length Φ(B) of a cycle basis of an unweighted graph is defined as Φ(B) := C B C. A minimum cycle basis (MCB) of a graph G is a cycle basis of G of minimum length. A set of circuits {C 1,...,C ν } such that C i \ (C 1 C i 1 ), i = 2,..., ν is clearly a cycle basis. We call such a basis weakly fundamental. Notice that these were already considered by Whitney [Whi35] in Let T be some spanning tree of G. Depending on the context, we either regard T as a subgraph of G or as a set of edges T E. For e E \ T, we denote by C T (e) or C e for short the fundamental circuit that e induces with respect to T, i.e., the unique circuit in T {e}. There are ν fundamental circuits associated with T. These form a cycle basis which is called strictly fundamental. Here, we may write Φ(T) instead of Φ(B). A minimum strictly fundamental cycle basis (MSFCB) has minimum length among the set of strictly fundamental cycle bases. In general, strictly fundamental cycle bases are a proper subset of weakly fundamental cycle bases, which in turn are a proper subset of general cycle bases of undirected graphs. In general none of the three corresponding minimization problems coincide [LR07].

51 2.3 Lower Bounds 45 At the same time, given a spanning tree T of G and any edge f T, the graph T f := T \ {f} is a forest comprising precisely two trees with vertex sets S f and S f respectively. We denote by δ(s f ) the set of edges in E with precisely one end-vertex in S f. This set δ(s f ) is called the fundamental cut of f with respect to T. There are n 1 fundamental cuts associated with T. These form a cut basis (or co-cycle basis) which is called strictly fundamental. We denote by Ψ(T) := f T δ(s f) the length of this strictly fundamental cut basis. With N N, the planar grid graph G N,N is the graph on V = {1,...,N} {1,...,N} with E = {{(i, j), (i, j )} i i + j j = 1} = {{u, v} u v 1 = 1}. In a graphical representation, e.g., in an embedding into Z 2, the first index of a vertex represents its x-coordinate, the second index its y-coordinate. Further define the vertical distance of an edge e = {(i, j), (i, j )} from a vertex u = (a, b) as dist V (e, u) := max{j, j, b} min{j, j, b}. The horizontal distance is defined analogously using the first coordinates i, i, a. The grid graph G N,N has n = N 2 vertices and contains m = 2 N (N 1) edges. Its cyclomatic number ν is (N 1) 2. During the whole chapter we refer to N as the dimension of the grid. We collect some well-known simple properties of the cycle space of such grids in the following proposition. Proposition 2.1. The planar grid graph G N,N has a unique minimum cycle basis B. In B each basic circuit contains precisely four edges, thus Φ(B) = 4ν = Θ(n). The basis B is weakly fundamental. But for N 4, B is not strictly fundamental. 2.3 Lower Bounds In this section we investigate lower bounds for the MSFCB problem on unweighted planar grid graphs. As mentioned in the introduction Alon et al. [AKPW95] proved that an optimal solution is of size Θ(nlog n). However, a detailed view on their presentation reveals that they achieve a lower bound of ln 2 2,048 nlog 2 n O(n). In Section we improve this lower bound by a factor of approx. 245 reaching a lower bound of 1 12 nlog 2 n O(n). Not surprisingly, this lower bounds only works out for very large dimensions. Thus, in Section we developed a combinatorial lower bound of 6.25n 23.5 n + 34 which outperforms the asymptotical bound for dimensions up to N = Further, we present two different mixed-integer linear programming approaches. In particular, in Section we introduce an approach by Amaldi et al. [ALMM04] for general graphs whereas in Section we develop a MIP that is purpose-built for grid graphs. We conclude the section on lower bounds with presenting a combinatorial approach for a tight lower bound for dimension N = 8 derived from ideas from Section

52 46 Strictly Fundamental Cycle Bases on Grids A New Asymptotical Lower Bound In this section we prove that an optimal solution to the MSFCB problem on a N N grid is of length at least 1 12 nlog 2 n O(n). The approach will make extensive use of the geometric dual of the grid. Therefore we begin with introducing some conceps that we will need. Consider the dual of an embedded planar graph G, which we denote by G. For a primal grid G N,N of dimension N N embedded into Z 2, there are (N 1) 2 finite faces, plus the infinite face F. Consider the finite face that is incident with the vertices (i, j), (i+1, j), (i+1, j + 1), (i, j + 1) of G N,N. We introduce a vertex in the dual graph G, and associate with it the coordinates (i, j), i.e., the coordinates of the bottom-left corner of this face. In this embedding, consider the subgraph G \ {F } of G that is induced by the dual vertices that we introduced for the primal finite faces. Here, the edge-face incidences in G N,N cause this subgraph to be a grid graph with (N 1) (N 1) vertices. Finally, the dual vertex F that we associate with the infinite face of G N,N is adjacent to all border-vertices of the dual grid G \ {F }. Note that for each of the four corner vertices (1, 1), (1, N 1), (N 1, 1), and (N 1, N 1) there exist two parallel edges with the other endpoint being F. Recall that the edge set of G can be identified with the edge set of G (see [Sch03, Ch. 3]). Consider a spanning tree T of G N,N and its dual counterpart, that we denote by T. In fact, T can be understood as the complement of T, as it contains the counterpart in G of each edge in E(G N,N ) \ T. The graph T is a spanning tree of G, although it is not necessarily connected when restricted to G \ {F }. The following key observation is well known (see [Sch03, Ch. 3]). There is a one-to-one correspondence between fundamental circuits w.r.t. T in G N,N and fundamental cuts w.r.t T in G. More precisely, F E(G N,N ) is a fundamental circuit w.r.t. T in G N,N if and only if F itself is a fundamental cut w.r.t T in G. Therefore, bounding sizes of cuts in the dual is the same as bounding sizes of circuits in the primal, in particular Φ(T) = Ψ(T ). In this section we first show that every strictly fundamental cycle basis B of the square N N grid with n = N 2 vertices satisfies Φ(B) 1 16 nlog 2 n O(n). Thus, our direct approach substantially improves the lower bound that has been obtained in [AKPW95, Thm. 6.6] by a factor of more than 245. Thereafter, we go one step 1 further and establish a lower bound of 12 nlog 2 n O(n). In contrast to [AKPW95] we decided to tackle the lower bound problem from the dual side, i.e., instead of the graph G itself we examine the geometric dual G of it. Some of the structural properties, e.g., as elaborated in Lemma 2.2, are easier recognized in this dual setting. For notational convenience, we only consider grids of certain dimensions. In particular the dual grids that we study have dimension N 1 = 2 k +1 (k Æ), implying V (G ) = N 2 2N +2. The corresponding primal grid is of size n = (2 k +2) 2. With this particular definition of N it is much easier to follow the recursive approach that is to be explained. Notice that also Alon et al. [AKPW95]

53 2.3 Lower Bounds 47 consider specific dimensions. The main ideas of our proof are as follows. We consider an arbitrary spanning tree T of the primal grid G N,N. Instead of counting the length of the strictly fundamental cycle basis that it induces, we look at the length Ψ(T ) of the strictly fundamental cut basis of its dual tree T. In several iterations which will be organized in levels we consider sub-paths of T that start at certain specified vertices of the dual grid. Each edge of these paths induces a fundamental cut. Yet, we consider only those fundamental cuts that are induced by specific subsets of the edges of these paths. We will denote these subsets as pseudo-paths. For one such cut, Lemma 2.2 provides a lower bound on its contribution to Ψ(T ). As pseudo-paths of different levels do in general intersect, in Corollary 2.6 we finally identify values that we may sum over all levels. As a first important tool we introduce pseudo-paths, the above mentioned subsets of paths. Consider two vertices u = (i, j) and v = (i, j ) in G \ {F } such that the unique u, v-path P in T does not contain F and i i. We now define a vertical and a horizontal pseudo-path, which exclusively consists of vertical and horizontal edges, respectively, that lead from u to v. More precisely, to obtain the horizontal pseudo-path P H u,v of P, we check whether i = i, in which case we set P H u,v =. Otherwise, starting from u we traverse the path P until we reach the first edge f with end-vertices w 1 = (i 1, j 1 ) and w 2 = (i 2, j 2 ) such that i = i 1, j 1 = j 2, and i 2 i = i 1 i 1. Now we recursively define the horizontal pseudo-path P H u,v as the following ordered set of edges, P H u,v := {f} P H w 2,v. We define the position of an edge f in P H u,v as { pos(f, Pu,v) H 1, if f = = f, pos(f, Pw H 2,v) + 1, otherwise, i.e., f Pw H 2,v. An equivalent procedure defines the vertical pseudo-path P V u,v. Observe that in general P H u,v P V u,v P; for an example of such a path P, in Figure 2.1(a) consider the two vertices with Cartesian coordinates u = (1, 1) and v = (2, 2). As an example for a pseudo-path, consider the dual graph T in Figure 2.1(a) and the black vertex u in the center of the grid. Let v be the penultimate vertex of the u, F -path. With this, P V u,v exactly consists of the black edges, highlighted in Figure 2.1(e). In addition, the position of an edge is tightly related to the quantity dist. Namely, pos(f, P V u,v) = dist V (f, u), for an edge f that is contained in a vertical pseudo-path starting at u. Lemma 2.2. Let u = (i, j) V (G ) \ {F } be some vertex in the dual grid and let v be a vertex on the (unique) path P between u and F in T. Further, let P u,v P be a pseudo-path between u and v. Then, the sizes of the fundamental cuts can be bounded by δ(s f ) 2 pos(f, P u,v ), f P u,v. (2.1) Proof. Without loss of generality, regard P u,v as a horizontal pseudo-path, and assume v = (i, j ) where i = i + P u,v.

54 48 Strictly Fundamental Cycle Bases on Grids Consider some edge f P u,v and the induced set S f V (G ) such that u S f and F S f, where δ(s f ) is the corresponding fundamental cut. As we consider a horizontal pseudo-path the y-coordinate of both vertices of f is equal and their x-coordinates are i + pos(f, P u,v ) 1 and i + pos(f, P u,v ), respectively. Remember that P u,v is contained in the unique u, F -path P of T. Therefore, all vertices between u and f are contained in S f. In particular, for each integer α with i α < i + pos(f, P u,v ), there exists a vertex in S f with α as x-coordinate. Out of those vertices in S f with x-coordinate α consider the vertex w max α with maximal (minimal) y-coordinate. Note that w max α to w max α (w min α (wα min ) and wα min may coincide. Now, ) one edge in the cut δ(s f ) can be assigned, because the dual vertex directly above (below) possibly F is not included in S f. Hence, for each α we get a contribution of two distinct edges and therefore a lower bound of 2 pos(f, P u,v ) on δ(s f ) in total. Remark 2.3. In the proof of Lemma 2.2 we detect special edges to be contained in the cut. However, all of those edges are vertical edges; recall that we assume w.l.o.g. that P u,v is a horizontal pseudo-path. Thus, one can add a +2 to the right hand side in (2.1) since every fundamental cut contains at least 2 edges of both adjustments, vertical and horizontal. On the other hand, this is of no effect for the n log n-coefficient. Note that in general (2.1) does not hold when choosing a vertex v which is not contained in the unique u, F -path in the dual tree. Furthermore, (2.1) does not hold either when considering all the edges of an ordinary path P instead of one of its two pseudo-paths. Moreover, the estimate in (2.1) can be far from being tight. Consider in Figure 2.1(f) the vertex u having Cartesian coordinates (16, 2). In a vertical pseudo-path that starts at u the first edge only contributes 2, although it induces a cut of length 18. In order to employ this powerful tool for estimating sizes of cuts, we need some more definitions. An important concept for our approach is the distance between two dual vertices. Let the grid graph G \ {F } be embedded in Z 2 as described in Section 2.2 and let u = (i, j) and v = (i, j ) be two vertices of it. Then the distance d u,v is defined as max{ i i, j j }, or u v. It is a simple observation that for any two distinct vertices u, v that are connected by a path in T \ {F } at least one of the two pseudo-paths from u to v has precisely d u,v edges. Next we assign tags to vertices in order to organize them in so-called levels. In a dual grid G \ {F } with (N 1) 2 = ( 2 k + 1 ) (2 k + 1 ) vertices we establish k different levels of vertices as follows. The level k only contains the unique grid s center vertex. The center-vertices of the four quarters of G \ {F } (which overlap on their borders) constitute level k 1. Recursively, each of these four quarters is again subdivided into four new quarters whose centers define the next levels. Hence, for 1 l k there exist 4 k l level-l vertices in G \ {F }. We further assign boxes to level-vertices. These boxes are exactly the quarters which were used to define their center-vertices as belonging to a certain level technically, for a vertex u of level l with u = (i, j) V (G ) \ {F }, we define

55 2.3 Lower Bounds 49 its box as the set of dual vertices B u = { v d u,v 2 l 1}. Further, we call the set { v du,v = 2 l 1} the border of B u. We illustrate the arrangement of the levels in Figure 2.1(a). There, the 64 = level-1 vertices are marked as small light-grey circles. With increasing level index the level-vertices are sketched with increasing intensity culminating with the one level-4 vertex in the grid s center. In Figure 2.1(b)-2.1(d), the boxes of the level-vertices are indicated by thin lines. In these figures levels 1, 2 and 3 are marked. In Figure 2.1(e), the box of the level-4 vertex constitutes the whole dual graph, except for F. We count along the following pseudo-paths. Every level-vertex u serves as a starting point of one pseudo-path. We consider the unique u, F -path P in G. Every such path has to intersect the border of box B u. Let v be the first such border vertex. For every level-vertex u we denote by P u,v the longer one of the two pseudopaths from u to v. Consequently, if u is a level-l vertex, then P u,v = 2 l 1 = d u,v. Sometimes, for a level-vertex u we may write P u instead of P u,v, as a shorthand. Lemma 2.2 suggests that we can count for every edge e P u a contribution of 2 pos(e, P u ) to the global lower bound. However, as pseudo-paths of different levels may overlap (cf. Figure 2.1(c)) this may over-estimate the lower bound. In fact, there even exist spanning trees such that one edge is contained in a pseudo-path of every single level. In the next two subsections we propose two strategies to overcome the inconvenience of overlapping pseudo-paths. The following two lemmas are the key observations to justify this approach. Proposition 2.4. Let u u be two level-vertices of levels l and l, respectively, such that their pseudo-paths P u and P u share some edge e. Then l l. Proof. The claim follows from two facts. First, every pseudo-path only consists of edges within its box. Second, boxes of the same level only intersect at their borders. Lemma 2.5. Let l, l be two levels. Let u u be two corresponding level-vertices such that their pseudo-paths P u and P u share some edge e. By Proposition 2.4, we may assume that l < l. Then pos(e, P u ) 2 pos(e, P u ). (2.2) Proof. Since u is a level-l vertex, we have pos(e, P u ) 2 l 1. Hence, it suffices to show that pos(e, P u ) pos(e, P u ) + 2 l 1. Denote by (i, j) the coordinates of u in the dual grid. Without loss of generality we assume P u to be a horizontal pseudo-path leaving its box B u at the eastern border, i.e., at x-coordinate i + 2 l 1. Then the endpoints of e have x-coordinates i + pos(e, P u ) 1 and i + pos(e, P u ), respectively. A simple but important observation is that the u-f -path and the u -F -path coincide from their first common vertex on, including the edge e. In particular, they traverse their common edges in the very same direction. But so do the pseudo-paths. Hence, P u is a horizontal pseudo-path leaving its box B u at its eastern border, too.

56 50 Strictly Fundamental Cycle Bases on Grids (a) A dual tree (b) Level 1 (c) Level 2 (d) Level 3 (e) Level 4 Figure 2.1: In (a) a dual tree T is sketched. The edges that overhang the grid indicate the connections to F. In (b)-(e) one can see which parts of T are used for bounding the corresponding cut-lengths in each level-iteration. Those are depicted using black lines, whereas the grey parts stand for pseudo-paths of previous levels. Note that the pseudo-paths do not necessarily start from the vertex they belong to. In addition, the boxes of the vertices of each iteration are illustrated. In (f) we illustrate what a small part of the tree is actually taken into consideration to obtain the desired lower bound. The bold edges are the ones that are shared by the level-4 pseudo-path and some pseudo-path of level 1, 2, 3. (f)

57 2.3 Lower Bounds 51 As e P u P u and l > l we obtain B u B u. On the one hand, by the definition of P u this path contains only edges with x-coordinates at least as large as those of the center vertex u of its box. On the other hand, since B u B u the pseudo-path P u contains 2 l 1 edges with both of their x-coordinates in the set {i 2 l 1,...,i} and thus is not contained in P u. Hence, pos(e, P u ) pos(e, P u ) + 2 l 1, which proves (2.2). By applying Lemma 2.2 inductively, we have the following result. Corollary 2.6. Let e be an edge which is contained in pseudo-paths P l 1,...,P l s of levels l 1,...,l s, where l s = max{l 1,...,l s }. Then s 1 pos(e, P l s ) pos(e, P l i ). (2.3) i=1 Now we continue with presenting two ways to deal with overlapping pseudo-paths. In a first approach, which then gives the 1 16 nlog 2 n O(n) lower bound, we voluntarily count less for each occurrence of an edge on some pseudo-path. This way, we may eventually sum over every occurrence of an edge on some pseudo-path. Thereafter, in an alternative refined analysis, we maintain some bookkeeping in which we keep track of all possible occurrences of the edges on pseudo-paths. So, recall from Lemma 2.2 that for every edge e twice its position on the path of the maximal level l s that e occurs on is a valid lower bound for δ(s e ). By Corollary 2.6 we have a valid lower bound when summing over every pseudo-path P l i that the edge e occurs on, its position values pos(e, P l i ), i.e., δ(s e ) (2.1) 2 pos(e, P l s ) (2.3) s pos(e, P l i ). (2.4) Theorem 2.7. Let G N,N be the planar grid graph with n = N 2 = (2 k +2) 2 vertices. For every spanning tree T of G N,N the size of the strictly fundamental cycle basis induced by T can be bounded from below by i= nlog 2 n + Θ(n). Proof. Let E(P) be the set of edges that appear on some pseudo-path corresponding to the dual tree T of T. Then, using Φ(T) = Ψ(T ), we conclude Φ(T) = δ(s e ) e E(T ) (2.4) e E(P) k l=1 δ(s e ) = Pu u level-l vertex k l=1 Pu u level-l vertex e P u pos(e, P u ). e Pu l max. level for e δ(s e )

58 52 Strictly Fundamental Cycle Bases on Grids Since on level l there exist 4 k l pseudo-paths of length 2 l 1 each, we finally conclude k 2 l 1 k ( ) Φ(T) 4 k l i 1 = 4 k l 8 4l + 2 l 2 l=1 = 1 8 4k k i=1 = 1 16 nlog 2 n + Θ(n). l=1 ( 4 k 2 k) = 1 8 (N 2)2 log 2 (N 2) + Θ(N 2 ) In the remainder of this section we perform a refined analysis using our concept of counting along pseudo-paths showing that, in fact, Φ(T) 1 nlog n + Θ(n). 12 In order to obtain a simple asymptotic proof of the nlog n lower bound as presented above we tried not to over-estimate contributions of edges. Now, we will show that we do not have to abandon the factor of 2 (cf. Lemma 2.2) as we did before in (2.4). Of course, this requires a more detailed examination of the occurrences of edges in different pseudo-paths. The approach in the sequel is as follows. For a given tree T we investigate occurrences of edges in pseudo-paths. Notice that we are interested in the highest level that an edge occurs on; see (2.4). However, for the analysis we also need low-level occurrences of an edge. We perform this level-examination using Algorithm 1. Nevertheless, the algorithm only does a bookkeeping job; the tricky part of the section will be the analysis of the output lb of the algorithm. Roughly speaking, Algorithm 1 works as follows. We iterate over the different levels, over the corresponding level-vertices, and over the edges on pseudo-paths that lie within such a level-vertex box. Then, in the core of the algorithm, Lines 5 10, it is checked whether an edge is contained in the pseudo-path P u of the box B u. If the answer is positive we store in λ(, ) a lower bound on the size of the cut induced by the current edge, cf. Lemma 2.2. If the answer is negative, the former lower bound is carried over, Line 8. For the upcoming analysis, we also keep a quantity µ(, ) that measures the marginal increase of λ(, ) when having incremented the level. Lemma 2.8. Let 1 l 0 k. Then after the l 0 -th iteration of Algorithm 1 δ(s e ) l 0 j=1 µ(e, j), for all e E(P). (2.5) Proof. First observe that by the definition of µ(e, l) in Algorithm 1 the sum in (2.5) is in fact a telescoping sum, and thus can be simplified to l 0 j=1 µ(e, j) = λ(e, l 0 ).

59 2.3 Lower Bounds 53 Algorithm 1: Compute lower bound on Ψ(T) input : A spanning tree T for G N,N. output: A lower bound lb on Ψ(T) init λ(e, 0) = 0, lb = 0 for l = 1,...,k do for all level-l vertices u do for all e B u E(P) do if e P u then λ(e, l) 2 pos(e, P u ) else λ(e, l) λ(e, l 1) µ(e, l) λ(e, l) λ(e, l 1) lb lb + µ(e, l) We know that for a level-j pseudo-path P j (j l 0 ) containing e, the value λ(e, l 0 ) is set to 2 pos(e, P j ). Thus the claim follows by Lemma 2.2. Corollary 2.9. Algorithm 1 computes a valid lower bound on the size of the fundamental cut basis induced by T. Proof. Lemma 2.8 proves that at the end of the algorithm lb is indeed a lower bound on Φ(T). We begin to develop the refined lower bound: e E\T δ(s e ) = = = e E(P) k l=1 k l=1 k l=1 δ(s e ) Lem.2.8 u level l vertex u level l vertex u level l vertex e E(P) k µ(e, l) l=1 e P u µ(e, l) (2.6) (λ(e, l) λ(e, l 1)) (2.7) e P u [( ) ( )] λ(e, l) λ(e, l 1) (2.8) e P u e P u Here, Equation (2.6) is indeed the result of re-ordering edges of pseudo-paths, taking into consideration that all summands with µ(, ) = 0 are disregarded. Recall that by the definition of the boxes, two pseudo-paths of the same level do not intersect. In (2.7) we used the definition of µ(e, l); see Line 9 of Algorithm 1. Now, with (2.8) where we just regroup the previous summation we continue as follows. Separately,

60 54 Strictly Fundamental Cycle Bases on Grids in Lemma 2.10 we give a lower bound on the positive term in (2.8) and identify in Lemma 2.11 an upper bound on the negative term in (2.8). Lemma Let u be a level-l vertex and let P u be its associated pseudo-path. Then e P u λ(e, l) = 1 4 4l + 2 l 1. Proof. Since e P u and because of Line 6 of Algorithm 1 we know that λ(e, l) = 2 pos(e, P u ). Therefore, e P u λ(e, l) = e P u 2 pos(e, P u ) = The last equality follows from the definition of the position of an edge on a pseudopath. Finally, a simple calculation concludes the proof of Lemma Now, we come to the more difficult part: obtaining an upper bound on the negative term in (2.8). 2 l 1 Lemma Let u be a level-l vertex and P u the according pseudo-path. Then e P u λ(e, l 1) l + 2 l (2.9) Proof. Without loss of generality assume that P u is a vertical pseudo-path that leaves its box via the northern border. Let NH(u) be the northern hemisphere of the box B u of u = (i, j), i.e., NH(u) := {v = (i, j ) d u,v 2 l 1, j j}. The second new definition introduces subsets of the northern hemisphere NH(u); see Figure 2.2 for an illustration. Let S l 1 = NH(v 1 ) NH(v 2 ) with v 1, v 2 NH(u) and both v 1, v 2 are level-(l 1) vertices. Further define sets S j for j = l 1,...,2 recursively as follows. S j 1 = NH(u) v is level (j 1)vertex, v / (S j S l 1 ) i=1 2i. NH(v) Geometrically, the S j can be seen as stripes within NH(u) lying one upon the other each time doubling their height, when incrementing j. Notice that according to their definition the S j do not fully partition NH(u), but rather leave one small stripe at the bottom of Figure 2.2. Namely, we have P u \ l 1 j=1 S j = 1.

61 2.3 Lower Bounds 55 S 4 S 3 S 2 S 1 u Figure 2.2: An illustration of the stripes S j, j = 1,..., 4, within the northern hemisphere NH(u) of the level-5 vertex u in G. Denote the edge in P u \ l 1 j=1 S j by e. As a first step towards the assertion of the lemma, regroup the occurrence of the edges of P u in the summation (2.9) according to the sets S j : e P u λ(e, l 1) = l 1 j=1 e S j P u λ(e, l 1). (2.10) Notice that λ(e, l 1) = 0 since the edge e was not contained in any pseudo-path of level l 1 or below. We make a simple observation. Fact Let u be a level-l vertex and P u its corresponding pseudo-path, which is a vertical pseudo-path that leaves its box via the northern border. Let e be an edge of P u with e P v for the pseudo-path of some level-j vertex v, j < l. Then e NH(v). Now consider an edge e S l 2 P u. Since e S l 2 we have e / S l 1, because by definition the sets of vertical edges that are induced by the stripes are distinct. Hence, we deduce that e / NH(v i ), i = 1, 2 for the two level-l 1 vertices v 1 and v 2 that are contained in NH(u). By Fact 2.12 we obtain e / P v1 and e / P v2. Thus, since we know e / P v1 and e / P v2 we also know that in iteration l 1 of Algorithm 1, λ(e, l 1) was not set in the if-statement (Line 6), but rather in the else-statement (Line 8). Thus, we know λ(e, l 1) = λ(e, l 2). In general, we observe that for an edge e S j P u we get λ(e, l 1) = λ(e, j), for all j = 1,...,l 1. Hence, λ(e, l 1) = λ(e, j). (2.11) e S j P u e S j P u Denote the 2 j 1 edges in S j P u from south to north by e 1, e 2,..., e 2 j 1. Then consider an edge e i and its corresponding value λ(e i, j) as in the right hand side of (2.11). Remember that we are still in one particular stripe S j. By Algorithm 1 there exists a level-j vertex v, j j, with λ(e i, j) = 2 pos(e i, P v ). Remember that e i P u which is a vertical pseudo-path, and thus P v is a vertical pseudo-path, too. Hence, by definition pos(e i, P v ) = dist V (e i, v ). Now let v be an arbitrary level-j vertex in S j. Notice that v lies on the southern boundary of S j. Then, dist V (e i, v ) dist V (e i, v), (2.12)

62 56 Strictly Fundamental Cycle Bases on Grids because v S j. By Fact 2.12 the edge e i P v cannot lie between v and v, because this would be south of v. However, since v is a level-j vertex, we have dist V (e i, v) 2 j 1. Now, consider the unique edge f i that lies in the pseudo-path P v of vertex v for which its position pos(f i, P v ) equals dist V (e i, v). Notice that P v may be a horizontal pseudo-path. In total, we deduce λ(e i, j) = 2 pos(e i, P v ) = 2 dist V (e i, v ) (2.12) 2 dist V (e i, v) = 2 pos(f i, P v ). (2.13) This naturally induces a function that maps an edge e i S j P u to an edge f i S j E(P) such that (2.13) holds. In fact, this mapping is injective. To see this, assume for two edges e i1, e i2 S j P u that f i1 = f i2. But then dist V (e i1, v) = dist V (e i2, v). However, because of e i1, e i2 P u we also observe pos V (e i1, P u ) = pos V (e i2, P u ). Hence, we deduce that in fact e i1 = e i2, because no two distinct edges of a pseudopath can have the same position value. Thus we have e i S j P u λ(e i, j) 2 j 1 i=1 2 pos(f i, P v ). (2.14) Finally, (2.10), (2.11) and (2.14) together with a simple calculation provide what we claim in Lemma 2.11 e P u λ(e, l 1) l 1 j=1 2 j 1 i=1 2 pos(f i, P v ) = l 1 j=1 2 j 1 i=1 2i = l + 2 l Now, we go back to (2.8) and substitute the results of the two previous lemmas. Hence, e E\T δ(s e ) = k l=1 u level l vertex k 4 k l l=1 [( ) ( l + 2 l l + 2 l 1 4 )] 3 ( 1 6 4l = 1 6 4k k + Θ(4 k ) ) (2.15) = 1 6 (N 2)2 log 2 (N 2) + Θ(N 2 ) = 1 12 n log 2 n + Θ(n) Notice that (2.15) holds because the bounds that we derived in Lemmas 2.10 and 2.11 only depend on the level, but not on the particular level-vertices. Hence, we get the following asymptotic lower bound.

63 2.3 Lower Bounds 57 Theorem Let G N,N be the planar grid graph with n = N 2 = (2 k +2) 2 vertices. For every spanning tree T of G N,N Φ(T) 1 12 n log 2 n + Θ(n). Remark In the asymptotic lower bound of Theorem 2.13 there are low-order terms hidden in the Θ(n). We mention two possibilities to increase these terms as well. First, in Lemma 2.2 we only looked at either horizontal or vertical edges. Nevertheless, an induced cut S f on a grid always contains both types. Hence, we actually could exploit δ(s f ) 2 pos(f, P u,v ) + 2 for an edge f in the pseudopath P u,v as already suggested in Remark 2.3. Second, while deriving the bound of Theorem 2.13 we only considered edges that lie in pseudo-paths. However, it can easily be checked that for the N N grid, with N = (2 k + 2), we have E(P) 1/2 (N 2 N). So, in Theorem 2.13 so far we do not consider at all the fundamental cuts of at least ν 1 2 (N2 N) edges. But since all fundamental cuts have size at least 4, we can increase our lower bound. In total, after including these three enhancements we identify as an even better lower bound. 1 6 (N 2)2 log 2 (N 2) N N The Challenge of Small Grids In the previous section we established an asymptotic lower bound on the size of a fundamental cycle basis. However, even when taking into account the considerations in Remark 2.14, it has to be mentioned that the trivial MCB bound is stronger in dimensions N 32. But these are important dimensions in which earlier empirical computations have been performed (e.g., [ALMM04], [LAM05]). Hence, for practically relevant dimensions even our improved asymptotic bound is only of partial use. Fortunately, in this section we can introduce a new better combinatorial lower bound which is 6.25n 23.5 n This will dominate the asymptotic bound in its form in Remark 2.14 not only for N 32, but rather up to N The main work to attain this lower bound is to first establish the slightly weaker lower bound of 6n 20 n And this work is presented mainly in the primal view on G N,N, with only a few times referring to predecessor edges in the rooted dual tree T. Let C be some circuit in G N,N. We denote by diam H (C) the horizontal diameter of C, i.e., the difference between minimum and maximum x-coordinates in Z 2 of vertices in C. Similarly, we consider the vertical diameter of C, denoted by diam V (C). It follows immediately that C 2 (diam H (C) + diam V (C)). (2.16) For a non-tree edge e T, we use diam H (e) := diam H (C T (e)) as a short hand.

64 58 Strictly Fundamental Cycle Bases on Grids e e diamv (e) diamv (e) diam H (e) (a) diam H (e) Figure 2.3: In the figure, the horizontal and the vertical diameter of the circuit induced by the non-tree edge e are illustrated. In addition, Inequality (2.16) is motivated. For the circuit in Fig. (a), (2.16) is a proper inequality. The thin edges can be detected as such that are counted in the left hand side of (2.16), but not in the right hand side. On the other hand, for the circuit depicted in Fig. (b), (2.16) holds with equality. (b) Let C be a circuit in G N,N and consider its enclosed finite region R. In C we collect all the edges in E(G N,N ) that are inside C. More precisely, these are those edges which are incident to two faces of G N,N that both have empty intersection with R 2 \ R. Proposition Let G M,N be the M N planar grid, let T be a spanning tree in it, and let and C be a simple circuit in G M,N. Then C T (e) 4 C \ T + 6 C \ T. (2.17) e (C C)\T Proof. Using (2.16) it suffices to show that 2 (diam H (e) + diam V (e)) 4 C \ T + 6 C \ T. e (C C)\T In order to show a lower bound for e (C C)\T (diam H(e)+diam V (e)) we will define in (2.20) a function d(e) having the property diam H (e) + diam V (e) d(e), for all e (C C) \ T. (2.18) Since e T we already know that diam H (e) 1 and diam V (e) 1, (2.19) implying that d(e) = 2 would already satisfy (2.18). Yet, to arrive at (2.17) we must increase d(e) beyond two. Consider the spanning tree T in the dual graph (G M,N ) that corresponds to E(G M,N ) \ T. Take F as the root of T. Consider the two

65 2.3 Lower Bounds 59 faces of G M,N that are incident with e. We refer to the one with the larger distance from F in T as F(e). Note that this face F(e) is inside the circuit C T (e). For each edge f (F(e) \ (C T {e}), denote by F(f) F(e) the other face that f is incident with. Observe that F(f) F because this would result in a circuit in the dual tree T. By the grid structure, each of these faces F(f) points into a different direction with respect to F(e), i.e., either north, east, south, or west. Now, since f T, we know that F(f) is inside C T (e). Therefore each edge f (F(e) \ (C T {e})) allows us to increment the lower bound on diam H + diam V, always satisfying (2.18). Formally, we set d(e) := 2 + F(e) \ (C T {e}). (2.20) Instead of summing up d(e) over the edges e (C C) \ T, we now rearrange the summation and count the contribution of each edge f C \T. By the definition of d(e) in (2.20) there are at most three edges f C \ T that contribute to the d( )- value of e. In turn, each such edge f is counted only for one edge e: its immediate predecessor on the dual path from F ; formally f (F(e) \ (C T {e})) = f / ( F(e ) \ (C T {e }) ) e e. So, we deduce To summarize, d(e) e (C C)\T e (C C)\T (2.20) = Finally, we conclude that e (C C)\T C T (e) F(e) \ (C T {e}) = C \ T. (2.21) e (C C)\T = 2 (C C) \ T + (2 + F(e) \ (C T {e}) ) e (C C)\T (2.21) = 2 (C C) \ T + C \ T F(e) \ (C T {e}) = 2 C \ T + 3 C \ T. (2.22) (2.16) (2.18),(2.20) e (C C)\T e (C C)\T 2 (diam H (e) + diam V (e)) 2 d(e) (2.22) = 4 C \ T + 6 C \ T. Corollary Let N 3 and let G N,N be the N N square planar grid with n = N 2 vertices. Then for each spanning tree T E Φ(T) = C T (e) 6 n 20 n (2.23) e E\T

66 60 Strictly Fundamental Cycle Bases on Grids Proof. Simply take C as the circuit that contains precisely the edges that are incident to F. Since E = C C, we apply Proposition 2.15 to C. There, we minimize the right hand side in (2.17) by maximizing C \T. Now consider the four vertices which are not incident to any edge in C. In any tree T, these must be incident to one edge in C T. As N 3, we conclude that C T 4, thus C \ T 4 n 8. Finally, a simple calculation yields (2.23). For N {3, 4, 5} there exist (primal) spanning trees that meet the (dual) bound of Corollary 2.16, and thus are optimum solutions for the MSFCB problem. At the end of this section we show how one can easily increase the bound in (2.23). To this end, recall from (2.20) that so far, to derive a bound on C T (e) for an edge e E \ T we were only considering local information: the tree-membership of three edges that share a face with the edge e. This way, the values d(e) were bounded from above by 5. However, in large grids it is easy to identify much larger circuits, e.g., when N is even, the ones that contain the innermost face in their interior. Taking some of these circuits, we obtain a slightly better bound. Theorem Let G N,N be the N N planar grid with n = N 2 vertices, N 10 and even. Then for each spanning tree T E Φ(T) = e E\T C T (e) 6.25 n 23.5 n (2.24) Proof. As we assume N to be even, the graph G N,N has a unique innermost face, which we refer to by F. Consider the pseudo-path P from F to F in the dual graph G. Applying Lemma 2.2 and Remark 2.3 to the edges (e 1,...,e N ) of P 2 provides C T (e i ) 2i + 2, i = 1,..., N 2. (2.25) Now, recall that in the bound in Corollary 2.16 each fundamental circuit was bounded from below by at most 10, because of d(e) (2.20) 5. In other words, taking for the edges e i, i = 5,..., N 2 the better bounds in (2.25) instead of d(e), lets us increase the lower bound in Corollary 2.16 by N 2 2(i 4). (2.26) Hence, (2.26) together with Corollary 2.16 proves Theorem i=5 Of course, further pseudo-paths can be considered. But this must be done with care, in order to prevent any inconvenience caused by overlapping paths, as experienced in Section Moreover, looking separately at horizontal and vertical diameters of the fundamental circuits further improves (2.24).

67 2.3 Lower Bounds The Amaldi MIP for the General MSFCB Problem For NP-hard combinatorial optimization problems often mixed-integer linear programming approaches are considered. This is also the case for the MSFCB problem for general graphs. Different MIP formulations have been proposed by Liberti et al. [LAM05] and Amaldi et al. [ALMM04]. In this section we qoute the MIP formulation by Amaldi et al. [ALMM04] for the MSFCB problem on general graphs. Moreover, we make it more efficient by identifying the first classes of valid inequalities. In particular, for planar grids two of these classes are even able to cut off any of the huge number of optimum solutions of the LP relaxation, thereby improving the lower bound of the root node in the Branch-and-Bound tree. Let G = (V, E) be a 2-connected graph with non-negative costs w e on an edge e E. To ensure a spanning tree T to be computed, we resort on the following characterization: T = V 1, and T is connected. We exploit the fact that T is connected, if and only if for each non-tree edge e = {i, j} E \T there exists a path in T between i and j. In their approach Amaldi et al. formulate the MSFCB problem as a multi commodity flow problem. In the following their mixed-integer linear program, see (2.27) is developed. Amaldi et al. introduce two sets of variables: first, binary variables z ij, for an edge {i, j} E, where z ij = 1 if and only if e T. For the definition of the second set of variables they introduce a directed version of G using the edge set A = {(i, j), (j, i) {i, j} E}. For each (i, j), (j, i) A they define w ij = w ji. Then, for each edge {k, l} E, the variable x kl ij 0 represents the flow through arc (i, j) A from source k to sink l. Observe that the constraints (2.27b) and (2.27c) manage the flow balance between k and l. Exactly one unit of flow has to be sent. The connection between the flow an the spanning tree is then given with Constraints (2.27d) and (2.27e). Namely, edges with strictly positive flow have to be in the tree. Notice that since fklow is sent between all k and l with {k, l} E Constraints immediately provides connectedness of the edge set defined by z variables. Finally, the Constraint (2.27f) ensures that actually a spanning tree is defined. In the Objective (2.27a), then, the first term collects the costs for all paths, associated with {k, l} E, and the costs of all tree chords. In the second term the cost of the tree branches, which erroneously was counted in the first term. Although the MIP formulation (2.27) has been observed to behave better than other formulations ([LAM05]), still there are some major shortcomings. First, the number of variables and constraints is large. For instance, there are 2 m 2 x- variables in other words Θ(N 4 ). Already with this simple observation one might not expect too much for the solvability with, say, N 20. Still, the second drawback is even worse. The LP relaxation has several trivial optimum solutions. For instance, take z N. This particular choice admits the x-variables to sum up to 4 (N 1) 2, being the optimum value of the minimum weakly fundamental cycle basis problem on G N,N. We will provide another set of optimum solutions of the LP relaxation in Example Of course one can check this to be the optimum value

68 62 Strictly Fundamental Cycle Bases on Grids of the LP relaxation by having a look at the dual problem. We conclude, adding valid inequalities to (2.27) will be key for its solvability. min {k,l} E (i,j) A w ij x kl ij + (x kl kj xkl jk j δ(k) j δ(i) (x kl {i,j} E (1 2z ij ) w ij (2.27a) ) = 1 {k, l} E (2.27b) ij x kl ji) = 0 {k, l} E, i V \ {k, l} (2.27c) {i,j} E x kl ij z ij {k, l} E, {i, j} E (2.27d) x kl ji z ij {k, l} E, {i, j} E (2.27e) z ij = n 1 (2.27f) x kl ij 0 {k, l} E, (i, j) A (2.27g) z ij {0, 1} {i, j} E. (2.27h) Thus, in the remainder of this chapter we provide three classes of valid inequalities: two, which are valid for general graphs and which are defined either in z- variables or in x-variables, and one class that exploits the particular structure of grid graphs and hereby can combine x- and z-variables. Lemma Consider the graph G = (V, E) and an arbitrary proper subset H of E. Denote by V (H) the vertices incident to edges in H. If V (H) V, then z ij V (H) z ij (2.28) {i,j} (V (H),V (H)) {i,j}/ H, i,j V (H) are valid for every integer feasible solution of the MIP. {i,j} H Proof. In an integer feasible solution the edges with z ij = 1 form a spanning tree T of G. Therefore, the right-hand side (RHS) of (2.28) equals the number of connected components of the graph with vertex set V (H) and edge set H. As we assume V \ V (H) to be nonempty, each component of (V (H), H) must be reachable from the former vertex set. Hence, to ensure the connectivity of G via edges for which z ij = 1, there must be at least as many such edges, as (V (H), H) has connected components. On the one hand, we are aware of graphs in which there exist fractional vectors (x, z), which are feasible for the LP relaxation and whose objective value is strictly smaller than that of an integer optimum solution. Hence, these inequalities could appear to be reasonable candidates to add to the LP-relaxation of MIP (2.27). On the other hand, it may happen that in an iterative cutting plane generation, one could only profit from inequalities of that type at rather late iterations.

69 2.3 Lower Bounds 63 Example Consider the grid graph G N,N. In Figures 2.4(a) and 2.4(b) we sketch two spanning trees T 1 and T 2 of G N,N. Denote the corresponding solutions of the MIP (2.27) by (x 1, z 1 ) and (x 2, z 2 ), respectively. (a) T 1 (b) T 2 (c) T 3 Figure 2.4: The Figures 2.4(a) and 2.4(b) depict the z vectors of the MIP solutions of two spanning trees T 1 and T 2 (in bold) for G 9,9. Figure 2.4(c) shows their convex combination z 3 = 1 2 z z 2 Now, consider the convex combination of (x 1, z 1 ) and (x 2, z 2 ) which results in the following fractional vector (x, z) := 1 2 (x 1, z 1 ) (x 2, z 2 ). Clearly, (x, z) is a feasible solution for the LP relaxation of MIP (2.27). But observe that there exists a vector (x, z) which is feasible for the LP, too, but in which {k,l} E (i,j) A { x kl 1, if zkl = 1 and ij = 2, otherwise. (2.29) In particular, the objective value of (x, z) equals 4 (N 1) 2. But this is just the optimum value of the LP relaxation of MIP (2.27). As z is the convex combination of two integer feasible points, any inequality that does not make use of any x-component, will never cut off (x, z), thus never increasing the LP value. From the above example we conclude that for planar grids no valid inequality having non-zero coefficients only in z-components, will ever be able to cut off one particular optimum solution of the LP relaxation, thereby never increasing the optimum value of any so refined LP. This is why in the sequel we investigate valid inequalities which are either defined purely in terms of x-variables, or as a combination of x- and z-variables. But before we do so, we prove a more general lemma that relates lengths of arbitrary cycles in graphs to distances in spanning trees of that graph. Therefore, denote by d T (e) the length of the unique path between the endpoints of e in a spanning tree T of an arbitrary graph G = (V, E). Lemma Let G = (V, E) be a 2-connected graph with a spanning tree T and consider a simple circuit C in G. Then, d T (e) 2 ( C 1). (2.30) e C

70 64 Strictly Fundamental Cycle Bases on Grids Proof. Let T denote an arbitrary but fixed spanning tree of G. In the following we will prove the claim by induction over C \ T. Notice first that C \ T = 0 would imply C T, which contradicts T being cycle-free. Therefore, we select C \ T = 1 as the inductive base. In this case, the distances d T (e) in the tree are one for the C 1 tree edges, and C 1 for the unique non-tree edge. Hence, the claim holds. In the inductive step, take a circuit C for which C \T = k 2. We will identify two circuits C 1 and C 2, each with 1 C i \T < k, i = 1, 2. Then, from (2.30) being true for C 1 and C 2 we argue that the claim is true for C. Consider two vertices u and v within C which are connected through a path P T such that V (P) V (C) = {u, v}, but E(P) E(C) =. Such a path exists because of C \ T 2, otherwise T would not be connected. Denoting by P 1 and P 2 the two paths between u and v defined by C, C 1 = P 1 P and C 2 = P 2 P are simple circuits in G. Now, as T is cycle-free, both P 1 and P 2 must include at least one non-tree edge, say e P1 and e P2. Otherwise T would have contained a cycle. But then, C 1 contains at least one non-tree edge less than the circuit C, because it omits P 2, thus e P2, and the path P contains only tree-edges. The very same holds for C 2. We may thus apply the inductive assumption to C 1 and C 2. Summing up (2.30) for these two circuits yields d T (e) + d T (e) 2 ( C 1 + C 2 2). e C 1 e C 2 By the construction of C 1 and C 2 it holds that C = C 1 + C 2 2 C 1 C 2, and thus d T (e) + 2 e C e C 1 C 2 d T (e) 2 ( C 1) + 4 C 1 C 2 2. (2.31) But since C 1 C 2 = P T we have that d T (e) = 1 for all e C 1 C 2 and Equation (2.31) simplifies to d T (e) 2 ( C 1) + 2 C 1 C 2 2. (2.32) e C Finally, because of C 1 C 2 1, and thus 2 C 1 C 2 2 0, Equation (2.32) implies (2.30) for the circuit C. Corollary Let G = (V, E) be a 2-connected graph and consider a simple circuit C in G. Then, 2 ( C 1) (2.33) e C f A is a valid inequality for every integer feasible solution of the MIP (2.27). x e f

71 2.3 Lower Bounds 65 Proof. Follows from Lemma 2.20 and the observation that for any integer feasible solution to (2.27) it holds that w a x e a d T (e), for all e E. (2.34) a A Note that the above corollary holds for general graphs. Yet, we are interested most in investigating its effect on grid graphs. When considering a grid graph, a smallest cycle has length 4. The following Lemma 2.22 generalizes Lemma 2.20 for such cycles. Notice that (2.35) is stronger than (2.17) in the sense that (2.17) with a LHS as in (2.35) brings a weaker bound, i.e., a smaller RHS. Lemma Let C be a cycle of length 4 of the grid graph G N,N. Then, a feasible integer solution of that graph must fulfill {k,l} C (i,j) A x kl ij 18 4 {k,l} C z kl. (2.35) Proof. A simple case distinction on the value of {k,l} C z kl, which is between 0 and 3, proves the claim. We refer to Fig. 2.5 for a discussion of the relevant cases (a) P z = 3 (b) P z = 2 (c) P z = 2 (d) P z = 1 (e) P z = 0 Figure 2.5: The cases for Lemma Tree edges are drawn in cyan, whereas for non-tree edges brown dotted lines were used. In green we indicate the extremal cases. That means, the green edges suggest tree edges outside the length 4 cycle C that, together with the cyan edges, constitute a subtree that minimizes the LHS of (2.35). For each edge of C the according flow value is depicted as well. To summarize, in Lemma 2.22 we gave a lower bound on the sum of flow values on a cycle of lentgh 4. Of course, one can also choose other cycles or general subgraphs to define cuts for (2.27). However, empirical studies showed that most of them are dominated by the cuts induced by the length 4 cycles, as presented im Lemma In the remainder of this section we report on experiments. For these experiments we consider grid graphs G N,N and try to find an MSFCB using the mixed-integer linear program (2.27) which we refined with the (N 1) 2 constraints of the type (2.35). We set the weights w to 1 for all arcs of the graph. In Table 2.1 we collect information on the MIPs and on their relaxation values.

72 66 Strictly Fundamental Cycle Bases on Grids Table 2.1: Some MIP stats and the relaxation values, such as the according CPU times, are depicted. We used CPLEX version 10.1 on an Intel P4 with 3.2 GHz and 1 GB RAM running Linux. Notice that for N = 5 the relaxation value of brings a lower bound of in fact 70. Additionally, information on the best known lower and upper bounds are given. N Best LP relax. MIP LB UB value time in s #Vars #Constraints #Non-zeros total binaries , ,177 17, , ,326 40, , ,181 80, , , , , , , , , , , , , , , ,896 Notice from Table 2.1 the huge number of variables and non-zero elements. Moreover, the time needed to solve the initial LP increases remarkably. However, observe that the actual values for the relaxation improve on the trivial ones by a MWFCB. Remember that using (2.27) without the additional cuts would have brought such trivial linear relaxations with value 4 (N 1) 2. Nevertheless, when using (2.27) and the additional cuts to optimally solve the MSFCB problem on G N,N we can only report of a moderate performance. Whereas for N = 5 it takes less than a minute, using CPLEX 10.1, to solve the MSFCB problem to optimality, already for N = 6 the running time of CPLEX increases to more than an hour. Of course we have to admit that Amaldi et al. built up their MIP (2.27) for general graphs and not particularly for grids. In the upcoming section we introduce a MIP that is especially tailored for the MSFCB problem on planar grid graphs A new MIP formulation In this section we introduce a new mixed-integer linear program formulation for the MSFCB problem. The MIP is specially tailored for grid graphs and we focus on finding good relaxations. Unlike in the MIP by Amaldi et al. that we described in the previous Section 2.3.3, we decided for a dual approach. In the following, we explain the variables, constraints and give a detailed interpretation. So, let G N,N be the embedded grid graph with n = N 2 vertices, see 2.2. We take the faces of the grid as the indexing set for the variables. Therefore, we identify a face with the coordinates of its upper left incident vertex. Define F := {(i, j) i, j = 1,..., N 1} as the set of faces and let (0, 0) denote the outer face F. Now, define

73 2.3 Lower Bounds 67 the following binary variables: and x (k,l) (i,j) = x (k,l) (i,j) = 1, if (k, l) lies on the unique dual path from (i, j) to F, 0 otherwise, 1, if (k, l) is the next vertex after (i, j) on the unique dual path from (i, j) to F, 0 otherwise. The variables x are defined for (i, j) {1,...N 1} 2, and (k, l) {1,...N 1} 2 {(0, 0)} with the condition that (i, j) (k, l). On the other hand, the variables x are defined for (i, j) {1,... N 1} 2, and (k, l) {1,...N 1} 2 {(0, 0)} that can be interpreted as neighbouring faces. The approach is to enforce the x and the x variables to define the dual of a spanning tree, or, more precisely, its incidence structure. Hereby, the x determine the actual tree whereas the x are needed to measure the length of cycles. First, we develop some constraints that enable the interpretation of a feasible solution of the MIP as a spanning tree. Afterwards we will motivate the objective function. Notice that the previous will be proven in Lemma min (i,j) F 2 (d HL (i,j) + d HR (i,j) dv T (i,j) + dv B (i,j) + 1) (2.36a) x (i,j) (k,l) + x(k,l) (i,j) 1 (i, j), (k, l) (2.36b) x (k,l) (i,j) + x(s,t) (k,l) 1 (k,l) (i,j), (k,l) x (0,0) (i,j) 1 (i, j), (2.36c) d HL (k,l) (i k) x (k,l) (i,j) (i, j), (k, l) (2.36d) d HR (k,l) (k i) x (k,l) (i,j) (i, j), (k, l) (2.36e) d V T (k,l) (j l) x (k,l) (i,j) (i, j), (k, l) (2.36f) d V B (k,l) (l j) x (k,l) (i,j) (i, j), (k, l) (2.36g) x(s,t) (i,j) (i, j), (k, l), (s, t) (2.36h) x (k,l) (i,j) x (k,l) (i,j) (i, j), (k, l) (2.36i) x (k,l) (i,j) = 1 (i, j) (2.36j) x (k,l) (i,j) = (N 1) (N 1) (2.36k) x, x binary First, we enforce anti-symmetry, that is, we give the incidence structure, which is to be established by the x variables, a direction. For example, if face (i, j) is

74 68 Strictly Fundamental Cycle Bases on Grids hanging below the face (k, l) on the same dual subtree rooted by (0, 0) then the opposite shall not be true. This is realized in (2.36b). Then we have to ensure transitivity of the incidence structure. Namely (2.36h) enforces that, if (i, j) is hanging below (k, l) and (k, l) is hanging below (s, t) on the same subtree, then (i, j) is also hanging below (s, t). Moreover, a direct incidence of faces has to imply a indirect incidence (2.36i). And as the number of tree edges in a spanning tree obviously is fix, the number of x variables set to 1 is also fix and in (2.36k) set to (N 1) 2. Furthermore, each face is hanging below the outer face, which is realized in (2.36c). And last but not least we enforce that each face has exactly one other direct predecessor on its path to the outer face, (2.36j). The evaluation of the length of a cycle, i.e., the objective, is motivated as follows. We introduce further variables d HL (horizontal, left), d HR (horizontal, right), d V T (vertical, top), d V B (vertical, bottom). These variables measure the horizontal and vertical diameter of the cycle. Namely, this works as follows. In a feasible integer solution, there is exactly one (k, l) with x (k,l) (i,j) = 1 for each face (i, j). Remember that this dual tree edge encodes a primal non-tree edge. Then, for the cycle induced by that particular non-tree edge, say e, the d variables measure the diameter. Namely, d HL (k,l) + dhr (k,l) + 1 equals diam H(e) and d V T (k,l) + dv B (k,l) + 1 equals diam V (e). This diameter counting is done in (2.36d) to (2.36g). Obviously it holds that twice the sum of the vertical and the horizontal diameter of a cycle is a lower bound on its length, cf. Equation (2.16) in Section However, there are two types of cycles, those that are evaluated correctly and those whose lengths are underestimated. See Figure 2.6 for an example. F (k, l) (k + 1, l) Figure 2.6: Here, an example is shown for a cycle whose length is underestimated by the objective of the MIP (2.36). In particular, the length of the cycle induced by the primal non-tree edge e = ((k, l), (k +1, l)) is approximated by (2.36d) to (2.36g) via d HL (k,l) = dhr (k,l) = 1, dv T (k,l) = 0 and dv B (k,l) = 2 with a value of 12. The actual length of the cycle, though, is 16. Now, we prove the following on a feasible solution of the MIP.

75 2.3 Lower Bounds 69 Lemma Every integer solution for the MIP corresponds to a unique spanning tree of the grid and vice versa. Proof. Consider a feasible solution, x and x for (2.36). Consider the dual G of the grid and herein the directed subgraph H defined by the edges {((i, j), (k, l)) x (k,l) (i,j) = 1 or x(i,j) (k,l) = 1}. It is sufficient to show that H is a spanning tree of G. Therefore, we show that H is cycle free. Together with the fix number of edges in H, see (2.36k), this implies the claim. First, notice that because of (2.36b) there are no cycles of length 2 in H. Further, assume that H contains a cycle of length greater than 2. Then, this cycle must be directed, since otherwise there is an (i, j) for which (2.36j) is violated. However, along this directed cycle one can now propagate, using (2.36i) and the transitivity (2.36h), a contradiction to the anti-symmetry stated in (2.36b). The statement of Lemma 2.23 together with the above mentioned fact that cycle lengths are either calculated correctly or underestimated, we deduce that every lower bound on the value of an optimal solution to the MIP (2.36) is a valid lower bound for the MSFCB problem on the considered grid graph. Table 2.2: Some MIP stats and the relaxation values as well as the according CPU times are depicted. We used CPLEX version 10.1 on an Intel P4 with 3.2GHz running Linux. For N = 13 the calculation in CPLEX was interrupted due to insufficient memory. Additionally, information on the best known lower and upper bounds are given. N Best LP relax. MIP LB UB value time in s #Vars #Constraints #Non-zeros total binaries ,077 11, ,547 45, ,580 1,436 46, , ,789 2, , , ,604 4, , , ,205 6, ,547 1,573, ,796 10, ,697 2,968, ,605 15,121 1,769,743 5,267,961 Similar as in the previous Section we now report on experiments evaluating the mixed-integer-linear program (2.36). In Table 2.2 we collect information about our studies. With respect to the relaxation value, the introduced MIP turns out to perform better than the Amaldi MIP, cf. Tables 2.1 and 2.2. In addition, the root relaxation solution of (2.36) is much faster than that of (2.27). On the other hand, the MIPs

76 70 Strictly Fundamental Cycle Bases on Grids become huge. More precisely, one faces a great number of binary variables, Θ(n 2 ) even if one keeps the x variables continuous, which maintains the MIP s validity. Moreover, the MIP contains a huge number of constraints. Namely, the transitivity constraints (2.36h) have to be defined for each triple of faces, that is Θ(n 3 ), or, Θ(N 6 ). Furthermore, when the task is to optimally solve the MIP (2.36) then it turns out that the crucial dimensions are N = 7 and N = 8. Whereas the MIP can be optimally solved for N = 7 in approx. 70s, for N = 8 more than nine and a half hours are needed. However, N = 8 herewith marks the largest grid for wich the MSFCB problem is optimally solved by any MIP approach. Notice that the obtained optimal solutions did not contain subtrees like in Figure 2.6. Therefore, these optimal solutions of (2.36) indeed constituted optimal solutions to the MSFCB problem. Actually, we think that these, in a sense, well-structered trees were not a coincidence. Rather, we conjecture that the optimal value of the MIP (2.36) indeed is the optimal value of the MSFCB problem. Conjecture For any optimal solution of the MIP, there is an equally valued spanning tree for which all induced circuits are evaluated correctly by the MIP. In other words, for any optimal solution T of the MSFCB problem on the grid G N,N there holds that e E(G N,N ) \ T the length of its induced cycle equals twice the sum of the cycle s diameters. Finally, a natural question that arises is, whether this MIP approach can be generalized to a broader class of graphs. However, since the approach uses the dual of a graph, other than planar graphs do surely not come into question. In fact, when considering planar graphs, the encoding of a spanning tree or of its dual, respectively, would work just as for grid graphs. Alone, counting lengths of cycles via diameters, as in (2.36d) to (2.36g), is not possible for general planar graphs, which makes a generalization a challenge A Tight Bound for G 8,8 We were able to solve the MSFCB problem on grid graphs having dimensions N 8: for N 5 through combinatorial arguments, cf. Section and especially Corollary 2.16, and for N = 6, 7, 8 through two different mixed-integer linear programming formulations, see [ALMM04] and Sections and It could be observed that the Machete-trees are optimal for dimensions N 7. However, at dimension N = 8 this structural property of an optimal solution is lost; compare the Machete trees in Figure 2.10 and the trees in Figures 2.9(a) and 2.9(b). Nevertheless, in G 8,8 the Machete-trees are still locally optimal with respect to exchanging one tree edge with a non-tree edge. This fact partly reflects the difficulties that a general analysis must overcome. Although we already solved the MSFCB problem on G 8,8, in this section we present a combinatorial proof for a lower bound, which, together with a known upper bound, solves the MSFCB problem for N = 8 to optimality, without using

77 2.3 Lower Bounds 71 any LP or IP theory. This combinatorial proof is built on top of some ideas of Section However, the limited consideration of rather local structures, see for example definition (2.20), will be enriched by a more global view on the dual of the spanning tree. The inner faces of the grid will play a key role to provide better estimates on the diameters of the circuits. Let T be a spanning tree of G 8,8 and T its dual. Moreover let C again denote the bordering circuit of G and C its interior. Observe that E = C C. We start by resuming from our analysis in Section 2.3.2: d(e) (2.20) = 2 + F(e) \ {C T {e}} (2.18) diam H (e) + diam V (e) During our investigation of G 8,8, we need to split d(e) into a horizontal plus a vertical part. More precisely, in the case of a horizontal edge e T, we define and d V (e) := 1 + (F(e) F O (e)) \ {C T {e}} (2.37) d H (e) := 1 + F(e) (F L (e) F R (e)) \ {C T {e}}. where F O (e), F L (e), F R (e) are the faces that share a border with F(e). See Figure 2.7 for an example of a horizontal edge where the F -path leaves F(e) southwards, and remember that we may also use a face to refer to its incident edges. Of course, we define d H (e) and d V (e) analogously for a vertical edge e. Observe that these F O (e)? F L (e) F(e)?? F R (e) F Figure 2.7: The relevant section around the non-tree edge e in G 8,8. Depending on which of the dashed edges are in the dual tree we set d, d V and d H accordingly. e definitions ensure d V (e) diam V (e) and d H (e) diam H (e). Recall that in Proposition 2.15, the computation of the lower bound actually made only use of d H and d V, which are limited to values of at most three and two, respectively. In particular, this cannot be tight for G 8,8. Hence, in the following definition we quantify what we were missing, H (e) := diam H (e) d H (e) and V (e) := diam V (e) d V (e). (2.38) But there is another point where our analysis that led to the 6 n 20 n + 22 bound possibly was not tight. In the proof of Corollary 2.16 we used a rough bound

78 72 Strictly Fundamental Cycle Bases on Grids on the number of tree edges on the grid s border. Later, in (2.41), we will also improve on this. The following series of inequalities illuminates the relation between the introduced quantities: e E\T C e (2.38) = = (2.23) = = 2(diam H (e) + diam V (e)) e E\T e E\T e E\T e E\T e E\T 2(d H (e) + H (e) + d V (e) + V (e)) 2( H (e) + V (e)) + 2(d H (e) + d V (e)) e E\T 2( H (e) + V (e)) + 4 C \ T + 6 C \ T 2( H (e) + V (e)) + 6 E \ T 2 C + 2 C T.(2.39) We may further subdivide the set C T. Let v i, i = 1,...,4, be the grid s four corner vertices. We know that each of them is incident with at least one edge e i in C T. For each of the four corner vertices v i, in (2.39) we count its corresponding edge e i directly, thus 2 ({e 1,..., e 4 }) T = 8. In the sequel, we only have to consider the set C := C \ {e 1,..., e 4 } in more detail: e E\T C e e E\T G 8,8 = ( H (e) + V (e)) + 6 E \ T 2 C C T e E\T 2( H (e) + V (e)) + 2 C T. (2.40) The main work then aims to provide good lower bounds for H (e) and V (e). However, this will not be sufficient for proving 262 to be the optimum value of the MSFCB problem on G 8,8. In addition, we must provide lower bounds on C T, and we do so by introducing values min (e) for e C \ T, min (e) := {f C T F(f) T (e)}, (2.41) where T (e) denotes all the faces that use the edge e on their path to F. Formally, for e C \ T we define min (e) := 0. Observe that each edge f C T is only counted for one edge e C \ T. Hence, e E\T min (e) C T. (2.42)

79 2.3 Lower Bounds 73 This way, we provide a lower bound on (2.40): e E\T C e (2.42) e E\T = = = e E\T e E\T e E\T 2( H (e) + V (e)) V (e) + e E\T e E\T min (e) 2( H (e) + min (e)). (2.43) ( V (e) + H (e) + min (e)) (e) (2.44) = (2.45) Now, by showing for an arbitrary spanning tree T that 16, we finally establish optimality of the trees depicted in Figures 2.9(a) and 2.9(b) our ultimate goal. In the remainder, we provide lower bounds on V (e), and on H (e) + min (e) for particular non-tree edges e. We start by bounding from below the value H (e)+ min (e) for some e C \ T. Lemma Let e C \ T be a non-tree edge on the border of G 8,8. Then H (e) + min (e) diam H (e) 1. Proof. According to (2.38) this lemma is proven if we show that min (e) d H (e) 1. We distinguish cases according to the value of d H (e). Obviously, if d H (e 1 ) 1 there is nothing to show. In the two cases that remain, for ease of notation we assume the edge e C \ T to be a horizontal edge on the southern border of G 8,8. So, consider now d H (e) = 2. W.l.o.g. the edge ẽ F L (e) F(e) is not contained in T; see Figure 2.8(a). Then, the edge e F L (e) C has to be in T, because otherwise the dual tree T would contain a circuit. Now, we distinguish two sub-cases. First, if e C we are done, because min (e) 1, c.f. the definition of min (e) in (2.41). Second, if e / C, by the definition of C the edge e must be incident to the grid s corner; see Figure 2.8(b). But then the other edge ê that is incident to the very same corner has to be in the tree, too, because otherwise we again detect a circuit in the dual tree T. Due to the definition of C we have ê C. Further notice that F(ê) T (e). Hence, min (e) 1 and we are done in the case of d H (e) = 2. Finally, assume d H (e) = 3, cf. Figure 2.8(c). Then there exist edges ẽ and ẽ with ẽ F L (e) F(e) and ẽ F R (e) F(e). Hence, the edges e F L (e) C and e F R (e) C are tree edges. By the definition of C and as we are in G 8,8 we know that at least one of these two edges e and e is in C. In case both are contained in C we observe min (e) 2 and are done. Otherwise, w.l.o.g. e / C. Then, e must again be incident to a corner vertex v. Moreover, the other edge ê incident to v is in T and in C. So, we observe e, ê C and F(e ), F(ê) T (e). Therefore, min (e) 2.

80 74 Strictly Fundamental Cycle Bases on Grids ẽ ê ẽ ẽ ẽ e e v e e e e e F F (a) d H (e) = 2 (b) The (c) d H (e) = 3 sub-case v = (1,1) Figure 2.8: The assumption of d H (e) = 2 (d H (e) = 3) directly implies e T (e, e T), because otherwise a circut in the dual tree exists. F To organize the proof of 16 conveniently, we introduce some further useful notation. Consider the nine innermost faces of G, {f 1,..., f 9 }, cf. Fig 2.9(c). Recall from Section that their influence on the diameters and thus circuit lengths can only be partly reflected in the values d V and d H, whence these innermost faces are of particular interest. For each tree T of G 8,8 and its dual counterpart T we know that e T if and only if e E \T. Denote by Pi the dual path in T from an innermost face f i to F. Let ẽ i be the last edge when traversing Pi from f i to F. Define the set Ẽ(T ) as the set of edges by which the nine innermost faces leave the dual grid, i.e. Ẽ(T ) := {ẽ i i = 1,...,9} and β := Ẽ(T ). Notice that 1 β 8, because some faces share their last edge on the path to F ; in particular the central face f 5 always shares its last edge with some other face. Further, after renaming we assume Ẽ(T ) = {e 1,...,e β }. Moreover, we consider the sub-trees of T that are induced by the last edge on the f i F paths. Namely for j = 1,...,β define Tj := Ps. s=1,...,9 ẽs=e j Observe that the dual sub-trees Tj are disjoint. To denote the number of f i-vertices that are contained in Tj we define γ j := T j {f 1,...,f 9 }, for j = 1,...,β. Further, for e E \ T recall (2.44) and let α(tj e Tj ) := (e) be the relative cost with respect to the number of innermost faces that T j contains. Proposition For an arbitrary spanning tree T of G (2.45) = e E\T γ j (e) 16.

81 2.3 Lower Bounds 75 Proof. Let T be an arbitrary spanning tree of G. Consider its dual sub-trees T j, as defined above. First we observe that if α(t j ) 2, j 1,...,β, then the proposition is proven, because then β j=1 e Tj (e) 9 2 = 18. Hence, in the sequel we only need to consider trees T in which there exists some j 0 1,...,β such that α(tj 0 ) < 2. (2.46) Let us take a closer look at such a sub-tree T j 0. Assume w.l.o.g. that T j 0 leaves G via the edge e 1 on its south border. In the edge sets L k := {f E f = {(i, k), (i + 1, k)}, i = 1,..., 7} for k = 1,...,5 we collect the horizontal edges of the same level, i.e., having the same vertical distance from e 1 (see Figure 2.9(c) for an illustration of L 3 L 5 ). To investigate T j 0, we will consider three cases, in which we profit from Lemma Case 1. {f 1, f 2, f 3 } Tj 0. In this case, we find a sub-path Pj 0 of Tj 0 that starts at e 1 and which traverses Level L 5. Then, by definition of the vertical diameter, we get diam V (e 1 ) 5. Moreover let e 2 be the first edge in L 2 when traversing Pj 0 from F and let e 3 be the first edge in L 3. Then, we observe diam V (e 2 ) 4 and diam V (e 3 ) 3 again trivially by definition. However, by definition (2.37) for all edges e we have d V (e) 2. Thus, 3 i=1 2 V (e i ) 12. Now, since α(tj 0 ) < 2 we can conclude that γ j0 7. But this implies immediately that diam H (e 1 ) 3. Applying Lemma 2.25 to e 1 yields 2 ( H (e 1 ) + min (e 1 )) 4. In total, we have 2( H (e 1 ) + min (e 1 )) V (e i ) 16. (2.47) Thus we are done with Case 1. Notice that (2.47), in fact, implies γ j0 = 9. Moreover, for an arbitrary sub-tree T j of T we showed i=1 ( {f1, f 2, f 3 } T j and γ j < 9 ) = α(t j ) 2. (2.48) Case 2. {f 1,...,f 6 } T j 0 = and {f 7, f 8, f 9 } T j 0. The intersection of T j 0 with L 3 provides diam V (e 1 ) 3 and 2 V (e 1 ) 2. So, α(t j 0 ) < 2 provides γ j0 {2, 3}. First, assume γ j0 = 3. Then diam H (e 1 ) 3 and using Lemma 2.25 we know that 2 H (e 1 ) + 2 min (e 1 ) 4. Therefore, 2( H (e 1 ) + min (e 1 ) + V (e 1 )) 6. However, since γ j0 = 3 this is a contradiction to (2.46). Second, if we assume γ j0 = 2, then we notice by our lemma that 2 H (e 1 ) + 2 min (e 1 ) 2. This leads to a very similar contradiction, because this would imply α(t j 0 ) = 2.

82 76 Strictly Fundamental Cycle Bases on Grids f 1 f 2 f 3 L 5 f 4 f 5 f 6 L 4 f 7 f 8 f 9 L 3 (a) A tree with length 262. (b) A different tree with length 262. e 1 (c) The innermost faces. Figure 2.9: Figures 2.9(a) and 2.9(b) show two optimal spanning trees. Notice that the tree in Figure 2.9(b) is a representative of the trees for which all inequalities in Case 1 hold with equality. Likewise, the tree in Figure 2.9(a) tightens the inequalities of Cases 2 and 3. Case 3. {f 1, f 2, f 3 } T j 0 = and {f 4, f 5, f 6 } T j 0. Hence, T j 0 has to intersect L 4. Then, by definition of the vertical diameter, one gets diam V (e 1 ) 4. Again, denote by e 2 the first edge in L 2 when traversing T j 0 from F. Then, diam V (e 2 ) 3. Thus, 2 V (e 1 )+2 V (e 2 ) 6. Again, by (2.46) we use α(t j 0 ) < 2 to deduce γ j0 4. But this implies that diam H (e 1 ) 2. Now, assume first that diam H (e 1 ) = 2. Lemma 2.25 lets us conclude that 2 ( H (e 1 )+ min (e 1 )) 2. But this implies = 8, and α(t j 0 ) < 2 forces γ j0 to be at least five. Together with {f 1, f 2, f 3 } T j 0 = this implies diam H (e 1 ) 3, a contradiction. Finally, Lemma 2.25 lets us conclude that 2( H (e 1 ) + min (e 1 ) + V (e 1 ) + V (e 2 )) 10, and hence γ j0 = 6. Similar to (2.48) we now know ( {f1, f 2, f 3 } T j = and {f 4, f 5, f 6 } T j and γ j < 6 ) = α(t j ) 2, (2.49) for an arbitrary sub-tree T j of T. So, what we have shown so far is the following. When we detect that T j 0 belongs to Case 1 we are done. If this is not the case, T j 0 has to belong to Case 3. Then we find 10 + j j 0 α(t j ) γ j. Now consider a tree Tj for some j j 0. Then, if Tj fulfills the condition of Case 1 we deduce by (2.48) that α(tj ) 2. Analogously, if T j belongs to Case 3 we use (2.49) to see α(tj ) 2. For the remaining Case 2, we also deduce α(t j ) 2. So, finally, since j j 0 γ j = 3 we have = 16. This concludes the proof of Proposition Now, the major work is done to obtain our final result:

83 2.4 Upper Bounds 77 Corollary The size of an MSFCB on G 8,8 is 262. Proof. This follows directly from (2.43) and Proposition 2.26 together with the tree in Fig 2.9(b). 2.4 Upper Bounds In this section we report on upper bounds on the MSFCB problem on grid graphs. The previously best known spanning trees are from Alon et al. [AKPW95]. They give trees with an induced cycle basis length of O(nlog n) with n beeing the number of vertices of the grid G N,N, i.e., n = N 2. They also showed that this is asymptotically best possible. However, we consider upper bounds of the form c nlog 2 n+o(nlog n) with a constant c. In Section we develop spanning trees that reach an upper bound of the above form with c = This improves on the constant that reach the trees of Alon et al. by a factor of more than four third. Not surprisingly, such asymptotic bounds only work out at very large dimensions since they make extensive use of recursive subtree structures. So, at the end of Section we propose adapted spanning trees that we show to be well-suited for dimensions N 100 in the experiments presented in Section 2.5. However, we start this upper bounds section by taking a look at the really small dimensions. For example, if one considers spanning trees for N = 3, 4, 5 an alleged structure is detectable. Even more, if ones checks such trees for dimensions N = 6, 7 the conjecture seems to approve. For N = 3,...,7 this specially structured Figure 2.10: Machete trees for dimensions N = 3,... 7 trees are depicted in Figure In [BKW03], Boksberger et al. introduce such trees as Machete trees and determine them to be optimal for the problem of finding Minimum Stretch Spanning Trees on unweighted square grids. In the UNTS, see Chapter 3 and Fig. 3.1, the Minimum Stretch Spanning Tree (MSST) Problem reads as (max, E, d T (u, v)). However, since the MSST problem and the MSFCB problem differ, see again Fig. 3.1, even on grids, it is not surprising that the Machete trees turn out to be not optimal for the MSFCB problem. Rather, as it is not hard to verify, their length on the grid graph G N,N, with N odd, is 4 2 N 3 2 i=1 i (2j + 2) + j=1 2 (2i + 2) = 1 3 N3 + 2N 2 13 ( 3 N + 2 = Θ N 1 i=1 n 3 2 ).

84 78 Strictly Fundamental Cycle Bases on Grids Thus, we go on by stating some easy to analyze asymptotical optimal trees. Therefore, consider grids of dimension N = 2 k, where k 1 is an integer. We define the spanning tree T N as follows. Take all the edges that are incident to the F face, except for one edge e = {u, v} where in one coordinate u and v have values N 2 and N 2 + 1, respectively. If N 4, partition the grid G N,N into four subgrids G N 2, N. 2 Apply recursion to these subgrids such that the missing edges of the four subgrids (a) (b) (c) Figure 2.11: (a) a representative of a family of trees that are asymptotically worst-possible; (b) a representative T of a family of trees that are only a constant factor away from the optimum; (c) a representative tree as it was used by Alon et al. [AKPW95] can be reached from either u or v along an angle-free either horizontal or vertical path. For N = 16, we illustrate the result of this procedure in Fig. 2.11(b). Observe that the fundamental circuit that is induced by some edge of one of the four subgrids exclusively consists of edges of this subgrid. Hence, we may make use of the recursive structure of T N when computing Φ(T N ). For N = 2, we have Φ(T 2 ) = 4 as the base of the recursion. Consider now the middle cross in the visualization of the tree in Fig. 2.11(b). The non-tree edges there are precisely the ones that are not contained in any of the four subgrids. We denote the total length of their fundamental circuits by f(n), which we must add in the recursive step. Hence, have to solve the following recursive function { 4, if N = 2, and Φ(T N ) = ( 4 Φ T N 2 ) + f(n) otherwise (i.e. N = 2 k, k 2). (2.50) To assess the value of f(n), in accordance with Fig. 2.11(b) we group the summation into five blocks: four on the vertical part of the middle cross (top-down), plus its horizontal part: f(n) = N 4 1 (2i + 2) + (3N 4 + 2i) + (3N 2 + 2i) + i=1 N 4 1 N 4 1 i=0 N 4 1 (4(N 1) + 2i) + 2 (2i + 2) = 13 4 N2 2N 6. i=0 N 2 1 i=1 i=0

85 2.4 Upper Bounds 79 Now, the recurrence from (2.50) is solved exactly by Φ(T N ) = 13 4 N2 log 2 N 15 4 N2 + 2N + 2 = Θ(nlog n). (2.51) Notice that, in the form of c nlog 2 n+o(nlog n) these trees reach a c = However, in the next section we are going to introduce trees that yield a constant c strictly smaller than 1. Nevertheless, we do not want to leave unmentioned the following: As already mentioned, Alon et al. [AKPW95] proved that optimal solutions to the MSFCB problem on square grids are of quality Θ(n log n). There, as for an upper bound, they considered a class of recursively defined spanning trees. We will refer to these trees by TAKPW N. See Figure 2.11(c) for an example. When analyzing the trees the authors developed that for their trees it holds Φ(TAKPW N ) = 2 nlog 2 n+o(nlog n). However, in [KLRW06] it could be shown that in fact Φ(TAKPW N ) = 4 3 nlog 2 n + o(nlog n). At the end of these general considerations on upper bounds, we would like to mention a simple observation. There are indeed strictly fundamental cycle bases that asymptotically meet the most general upper bound for any cycle basis, ν n, thus Θ(n 2 ) in our case. As an example, we refer to the double snake tree in Fig. 2.11(a) A New Asymptotical Upper Bound Although Alon, Karp, Peleg, and West ([AKPW95]) think of their trees as being essentially optimal, we are able to construct trees with an asymptotic coefficient for the nlog 2 n term being strictly smaller than one. Namely, we provide spanning trees that induce strictly fundamental bases of size not more than nlog 2 n + o(nlog 2 n). These spanning trees will be defined for large dimensions. However, at the end of this section we present similar trees which empirically perform very well already in small dimensions. Then, we make use of this in the experiments Section 2.5. Before we describe how we construct the asymptotically good spanning trees, with the next paragraph we motivate how a class of recursively defined trees looks like that, in fact, accommodates both of the above mentioned goals. These spanning trees are the union of spanning trees in rectangular subgraphs of G N,N, their building blocks. The trees differ in how their rectangular subgraphs all respecting some arbitrary but fixed aspect ratio α 1 partition the faces of G N,N. Hence, it remains to specify how to construct a spanning tree subject to a given parameter α for some grid G M,N having aspect ratio max{ M N, N M } α. This is done recursively. Assume w.l.o.g. that M N. At the top-level of the recursion, we add to T α (G M,N ) the edges of the two longer borders of G M,N (here the horizontal ones), plus of one of its two other borders (cf. Figure 2.12). For the recursion, we partition the faces of G M,N into almost equally-sized rectangular subgraphs of aspect ratio again being close to α; only the faces of one horizontal path in (G M,N ), located almost in the middle of its two horizontal borders, are not contained in any of these rectangular subgraphs.

86 80 Strictly Fundamental Cycle Bases on Grids sub-sub-block sub-block Figure 2.12: The shape of a block (left) and with a sketched interior recursively filled with smaller blocks (right), always keeping the aspect ratio. These trees are related to other families of trees as follows. In G N,N, choosing α N 2 : 1 there exists a partition of the grid such that we end with Machetetrees ([Bok03] and [BKW03], cf. Figure 2.10). Moreover, an aspect ratio of α = 1 : 1 yields trees which can be obtained alternatively by a construction that is much similar to the one for T AKPW. According to the requirement asymptotical or empirical quality one can consider trees with a block structure either having an aspect ratio of approximately 3 : 1 or an aspect ratio of 2 : 1, respectively. In addition, the trees differ in how the blocks are actually used to define a tree. Whereas on large grids it is sufficient to cover the grid with three (almost) equally-sized 3 : 1 blocks, for small dimensions the grids are tiled with many 2 : 1 blocks of many different sizes. To achieve a good asymptotical upper bound we decided to construct trees out of the above described blocks with an aspect ratio of 3 : 1. Unfortunately, it turns out to be tricky to subdivide or tile a square grid of arbitrary dimension with these particular blocks. Thus, we construct our trees bottom-up like. That means we take an atomic block of size 6 14 and arrange 32 copies of such a block to a new one having size This procedure is then iterated providing spanning trees for dimensions ( k/ ) ( k/ ) 31 (2.52) with k chosen integral and even. Finally, three copies of such a tree can be put onto each other and cover the entire square grid. Now, a detailed description of the construction of the tree and a precise analysis of it follow. Construction of the tree. Whereas before we gave a brief top-down description of the tree that we consider we now introduce them bottom-up like, thereby having more control on the dimensions and, thus, by-passing rounding indispositions.

87 2.4 Upper Bounds 81 For every k Æ + we construct recursively spanning trees as follows. For k equal to 1 consider the spanning tree T 1 as sketched in Figure This tree is defined on a 6 14 grid and it has its exit on the lower horizontal border. The next tree, T 2, is constructed by arranging 32 copies of T 1. First, 16 copies are glued with this particular orientation side by side. Second, we mirror the other 16 copies of T 1 horizontally and place them such that their exits are opposite to the first 16 copies e T1 Figure 2.13: The spanning tree T 1 out of which all the trees T k are constructed. T 1 has dimension 6 14, or side-lengths 5 13, respectively. At last, one vertical edge, which we will call e T2 is added to connect the two soconstructed connected components. The general rule here is to take the left vertical edge as connecting edge for the construction of the tree T k with k even and the upper horizontal edge for the construction of the T k with k odd. See Figure 2.14 for an example. By this construction, the tree T 2 is of dimension In general, the tree T k is constructed out of 32 copies of T k 1 and an additional connecting edge the very same way. In order to finally state a spanning tree for a square grid and to prepare the analysis of the trees we introduce four sequences for the x and y length, w.r.t. number of edges, of T k in dependence of k. As T 1 is a 6 14 grid tree we have x 1 = 5 and y 1 = 13. By construction, we get the following sequences taking the parity of k into account: x 2i = 16 x 2i 1 y 2i = 2 y 2i (2.53) for trees T k with k = 2i even. For odd k = 2i + 1 the tree T k has dimension x 2i+1 = 2 x 2i + 1 y 2i+1 = 16y 2i. (2.54) In the following we will only consider the spanning trees T k for k even. Notice that the T k with k even, always have their exit on the left vertical border of the grid. Now, simple calculations transform (2.53) and the start values x 2 = 80 and y 2 = 27, respectively, into the explicit sequence of the side-lengths of T k for even k: x k = k/ y k = k/ (2.55) If we now take a closer look at T k, k even, we see that the ratio of its lengths is almost 3 1. In fact, the exact ratio x k to y k is always greater than 2.96 and converges to Hence, if we take three copies of T k and put them one upon another, then the resulting spanning tree, let us denote it by Tk 3, covers a grid of dimension ( k/ ) ( k/ ). 31

88 82 Strictly Fundamental Cycle Bases on Grids We now claim that three times the size of T k is an upper bound on the size of Tk 3 restricted to the square grid G with dimension ( k/ ) ( k/ ). 31 So, how do we restricted Tk 3 to a square of the above size? Let us consider the boundary line L of G that, in a sense, cuts through the down-most copy T k of Tk 3, cf. Figure This T k consists of several subtrees T k 1, T k 3,..., T 1. Those odd subtrees can have their exit pointing downwards, T j, or upwards,denoted by T j. If for a j = 1, 3,...,k 3, k 1 the boundary line L cuts through a subtree T j we leave this part of tree unchanged and simply cut away what overhangs L. In the other case where the boundary line L cuts through a subtree T j we cut away the overhanging parts as well, but since we loose connectivity we add an edge to T j exactly where formerly the exit had been. If we do so for all j = 1, 3,...,k 3, k 1 we finally come up with a tree, let us denote it with T k whose chords induce cycles with lengths not greater than they had been before, i.e., within this down-most copy T k of Tk 3. Now, we continue with the analysis of T k. T k copies... f 2 f 2 f 2 T k 1 f 1... T k 1... L Figure 2.14: A schematic illustration of the tree T k for an even k Æ +. Due to the construction rules T k consists of 32 2 = 1024 copies of T k 2 and different slots, i.e. one main slot f 1, dark-gray, and 32 subslots f 2, light-gray. Analysis of the tree. So, for a k Æ + and k even, consider the spanning tree T k of the square grid with dimensions ( k/ ) ( k/ ). 31

89 2.4 Upper Bounds 83 We are interested in an upper bound on the strictly fundamental cycle basis induced by T k. As argued above we have Φ( T k ) 3 Φ(T k ). (2.56) In the following we develop a recursive formula for Φ(T k ). Because of the tree s special construction the following recursive formula holds, Φ(T k ) = 1024 Φ(T k 2 ) + f(t k ), (2.57) where f(t k ) denotes the size of the fundamental cycles induced by edges that do not lie entirely within a copy of the smaller T k 2 tree. We call those areas slots. Then, f(t k ) can be canonically subdivided into one main-slot and several sub-slots, cf. Figure Obviously, f(t k ) = f 1 (T k ) + 32 f 2 (T k ) (2.58) holds. Then, with the help of the sequences defining the lengths of the trees (Equations (2.53) and (2.54)) we straight-forward express f 1 and f 2 as f 1 (T k ) and 1 32 x k+1 i=1 f 2 (T k ) 2i + 15 j= y k 1+1 i=1 2 2i x k+1 i=1 15 j=1 + (2y k + 2j 16 x k + 2i) y k 1+1 i= y k 1+1 i= x k+1 i=1 (2x k + 2y k + 2i), (2.59) (2x k 1 + 2j 16 y k 1 + 2i) (2.60) (2x k 1 + 2y k 1 + 2i), (2.61) respectively. Further, plugging (2.59) and (2.60) into (2.58) and then (2.58) into (2.57) we yield the recursion: Φ(T k ) 1024 Φ(T k 2 ) + 20,323, ,064 32k + o(32 k ). (2.62) Here, we omit the value for the recursion start T 2 because it is of no importance for the coefficient of the n log 2 n term. After resolving (2.62) and applying the result to (2.56) one gets Φ( T k ) 60,970,059 1,968,128 32k k + o(32 k k). Finally, making use of the special dimension, i.e., n = k/ ,

90 84 Strictly Fundamental Cycle Bases on Grids one can state the following upper bound: Φ( T k ) 6, 774, 451 6, 922, 240 n log 2 n + o(n log 2 n). We summarize the paragraph on the new asymptotical upper bound by stating the following lemma: Lemma Let G N,N denote the N N square planar grid with n = N 2 vertices and with N = k/ for some even integer k. Then the size of a minimum strictly fundamental cycle bases on G N,N can be bounded from above by n log 2 n + O(n). Remember that thereby the previously best asymptotical upper bound by Alon et al. ([AKPW95]) is enhanced by a factor of more than four third. Now, as mentioned before the description of the asymptotical good spanning trees, we consider similar trees that induce short SFCB already for small dimension. The 3 : 1 block structured trees, as described in the above paragraph are not perfectly suited for smaller dimensions. As shown, 3 : 1 is asymptotically a very good aspect ratio. Yet, it is not possible to decompose an arbitrary square grid into 3 : 1 blocks without losing much of their advantage because of rounding errors. Therefore, for small grids, we chose a different block-structured graph. This time we use an aspect ratio of 2 : 1. In contrast to the above, the 2 : 1 blocks, do not really cover, but rather tile the square grid. The tiling procedure roughly goes as follows: At first, two opposite 2 : 1 blocks are put in the middle of the grid. See for example the two blocks marked with A, having side lengths 8 15 in Figure Next, horizontal 2 : 1 blocks (marked with B ) are added centrally aside such that rectangular subgrids in the four corners remain. In those corners (marked C ) we always direct the next block such that its depth can be chosen as small as possible, while its aspect ratio should stay as close as possible to the target ratio 2 : 1. During this procedure we do not need to pay attention to any rounding inaccuracies. In Figure 2.15 an example 2 : 1 block structured tree for dimension N = 31 is shown. The empirical quality of the so defined trees for small grids will be evaluated in the next section. 2.5 Experiments In this section we compare different spanning trees with respect to the length of the strictly fundamental cycle basis they induce. The experiments conducted in this section are intended to provide benchmark results considering both upper and lower bounds for dimensions N = 5,...,100.

91 2.5 Experiments 85 C B B B C A A C B B B C Figure 2.15: Notice the parquet-like structure of the tree with tiles having heightwidth ratio of 2 with small errors due to roundings. Inside, the blocks themselves are recursively filled with smaller blocks still maintaining the 2 : 1 ratio. In addition to the degree-based tree-growing heuristics that we already referred to in the introduction of this chapter (Section 2.1), local search techniques are considered. For this local search approaches, the neighborhood of a spanning tree is defined as follows. Let T be a spanning tree of an arbitrary graph G. Then an edge e T induces a fundamental cut in G with respect to T. Let f be an edge of that cut. Then, T := T {e} + {f} constitutes a spanning tree. Such a exchange of edges is called edge swap. See Fig. 2.16(a) for an illustration. However, this neighborhood (a) Figure 2.16: In Figure (a) an edge swap is illustrated: the red edge induces a fundamental cut out of which the green edge is taken to anew constitute a spanning tree. In this example, the edge swap decreases the size of the induced strictly fundamental basis. However, one may get stuck when using edge swaps to improve the basis. This is shown in (b). There, although the tree is locally optimal, the blue highlighted subtree causes the non global optimality. (b) is not exact. That means, see Figure 2.16(b) for example, there are local optimum trees, which are not globally optimal. Amaldi et al. ([ALMM04]) reported the performance of several strategies for searching the neighborhood of a spanning tree. In what they denote by local search (LS),

92 86 Strictly Fundamental Cycle Bases on Grids the entire neighborhood is examined and they move to the tree with the best improvement. In a second deterministic strategy (ES, for local search with edge sampling) only a restricted number of neighbors are tested. To prevent LS to terminate too early in a too bad local optimum, Amaldi et al. ([ALMM04]) run metaheuristics such as variable-neighborhood search (VNS) and a tabu search (TS) on top of LS. In any of their computations, an adapted version of the tree-growing heuristic of [Pat69] is used as the initial solution. In our computations, we use the 2 : 1 block-structured tree as initial solution. In contrast to (LS) we do not examine the entire neighborhood for improvement. Instead, whenever we identify a neighbor that improves the current solution, we greedily move to that neighbor. Of course, this method depends on the order in which the edges in the tree are checked. Empirical studies showed, however, that the influence of the edge-order is neglectable. For our computational studies we chose a random order of edges and ran our greedy-like approach denoted by (GS) ten times, considering the best value of the length of the cycle basis and the according running time of (GS). We skip average values, because we see the goal of the study in giving benchmark results. Examining the quality of the heuristic is only a secondary goal. Among the ten sample runs the lengths of the cycle bases vary by less than 1% only, anyway. In Table 2.3 we compare the constructive heuristics, i.e., those that build up a tree without doing any subsequent local improvements. Moreover, we complement these values with information on lower bounds obtained by Corollary 2.16, for odd dimension, and by Theorem for dimensions N 10 and N even, as well as with values of a minimum cycle basis. The latter were also used in the recent study of Amaldi et al. [ALMM04]. Note further, that N = 130 is the dimension closest to 100 for which our asymptotic lower bound, see Remark 2.14 is defined exactly. We mention that for this value of N the bound that we derived in Theorem 2.17 is by more than 40% stronger than the asymptotic bound. In our tables the italic numbers highlight the best known upper and lower bounds. For N = 5, these coincide and we mark this in boldface. Observe that for any dimension N 10, the new trees that we proposed at the end of Section yield smaller SFCB values than any of the other constructive heuristics. In Table 2.4 we compare the different local-search-type heuristics. For our greedy search (GS) we used a 3.2GHz Intel P4 computer ( A1 ), running Linux and using LEDA c. Amaldi et al. used for their local search heuristics (LS) and (ES) also an Intel P4 computer running Linux, but with 2.66GHz ( A2 ). Accordingly, the times stated in Table 2.4 refer to the particular architecture. The values for the metaheuristics (TS) and (VNS) also quoted from [ALMM04] each refer to 10 minute runs on the A2 environment. Much similar as in the purely constructive context, our new solutions improve the best known upper bounds for all dimensions N 20. As already mentioned before we ran our local search (GS) with a random order of the edges. In Table 2.4 the first two columns present the value for the best run out of ten samples, and the according running time, respectively.

93 2.6 Conclusions and Open Questions 87 Table 2.3: Comparison of the cost of some selected trees, i.e., the length of the according strictly fundamental cycle bases. The rightmost column presents the previously best lower bound for small dimensions, obtained just by 4 (N 1) 2. The penultimate column now states the consistently better lower bounds due to Corollary 2.16 and Theorem N 2 : 1 AKPW Machete C-Order Deo s NT Cor MWFCB [AKPW95] [Bok03] [LAM05] [LAM05] and and [DKP95] Thm However, it has to be mentioned that only for dimensions N {60, 80, 90, 100} the start tree had not already been locally optimal. 2.6 Conclusions and Open Questions In this chapter we investigated the Minimum Strictly Fundamental Cycle Bases (MS- FCB) problem on grid graphs. First, we gave a new proof of an Ω(nlog n) asymptotical bound on the size of an optimal basis that uses dual graphs and a new concept of pseudo-paths. A detailed analysis revealed that, w.r.t. the factor before the nlog 2 n, we enhanced the previously best lower bound by a factor of more than 245. Then we switched to small dimensions where we combinatorially developed a lower bound which, although asymptotically not optimal, turns out to be the best lower bound for dimensions up to N = Moreover we considered two different MIP formulations and established new cuts for them. Then, for dimension N = 8, we identified a tight lower bound hereby for the first time proving 262 to be the optimal value of the MSFCB problem for G 8,8. As for the asymptotical upper bounds, we developed a family of spanning

94 88 Strictly Fundamental Cycle Bases on Grids Table 2.4: An overview of the quality of five local search approaches. Missing values are marked with an and running times are measured in mm:ss. The columns (LS) (TS) are cited from [ALMM04]. N (GS) (LS) (ES) (VNS) (TS) cost time cost time cost time cost cost : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :01 trees improving on the previosly best known ones by a factor of more than 4 3. And, last but not least, we conducted some experiments that establish benchmark results. Next, we list some open questions. A first naturally question is, whether the gap between the lower bound and the upper bound on an MSFCB on a grid can be further reduced. Then, concerning mixed-integer linear programs for the MSFCB problem it remains to build specially tailored formulations for planar graphs. Moreover, the complexity status for the MSFCB problem remains open for grid graphs and planar graphs. We conclude this chapter with depicting spanning trees, see Figs and 2.18, which constitute best known upper bounds for dimensions N = 9,..., 20. With the exception of the tree for N = 20, which was provided by [DK08], these trees were found by hand, rather than by a certain heuristic.

95 2.6 Conclusions and Open Questions 89 (a) N = 9: 356 (b) N = 10: 466 (c) N = 11: 592 (d) N = 12: 734 (e) N = 13: 894 (f) N = 14: 1072 Figure 2.17: Best known spanning trees for dimension 9 to 14.

96 90 Strictly Fundamental Cycle Bases on Grids (a) N = 15: 1268 (1276) (b) N = 16: 1486 (c) N = 17: 1720 (d) N = 18: 1972 (e) N = 19: 2244 (f) N = 20: 2534 Figure 2.18: Best known spanning trees for dimension 15 to 20.

97 3 CLASSIFICATION OF TREE SPANNER PROBLEMS In this chapter we deal with tree spanner problems. Tree spanner problems have important applications in network design, e.g. in the telecommunications industry. Mathematically, there have been considered quite a number of max-stretch tree spanner problems and of average stretch tree spanner problems. We propose a unified notation for 20 tree spanner problems, which we investigate for graphs with general positive weights, with metric weights, and with unit weights. This covers several prominent problems of combinatorial optimization. Having this notation at hand, we can clearly identify which problems coincide. In the case of unweighted graphs, the formally 20 problems collapse to only five different problems. Moreover, our systematic notation for tree spanner problems enables us to identify a tree spanner problem whose complexity status has not been solved so far. We are able to provide an NP-hardness proof. Furthermore, due to our new notation of tree spanner problems, we are able to detect that an inapproximability result that is due to Galbiati [Gal01, Gal03] in fact applies to the classical max-stretch tree spanner problem. This chapter is based on [LW07]. 3.1 Introduction We consider a weighted connected undirected graph (G, w), where G = (V, E). We assume the edge weights to be positive integers, occasionally after scaling. Let T be a spanning tree of G. Depending on the context, we think of T either as a subset of the edges of G, or as a subgraph of G. For a spanning subgraph H of G and u, v V we denote by d H (u, v) the length of a shortest (u, v)-path in H. 91

98 92 Classification of Tree Spanner Problems In [CC95] the t-tree spanner problem has been introduced as follows: Decide whether there exists a t-tree spanner, i.e., a spanning tree T of G such that d T (u, v) t, (u, v) V V := V V \ {(v, v) v V }. (3.1) d G (u, v) The corresponding optimization problem of constructing a spanning tree that realizes the minimum value t among all spanning trees is called the Minimum Max-Stretch Spanning Tree (MMST) problem ([EP04]). Applications of the MMST problem arise in the area of network design, e.g. in the telecommunications industry. There, trees are of particular interest, because they allow to keep the routing protocols simple ([HL02]). In [PT01] the related problem of finding a Minimum Average-Stretch Spanning Tree (MAST) has been considered: Let w = 1, i.e., G is an unweighted graph, find a spanning tree T that minimizes {u,v} E\T d T (u, v) d G (u, v). (3.2) Since for unweighted graphs there holds d G (u, v) = 1 for all {u, v} E, it is a simple observation ([AKPW95]) that this MAST problem turns out to be nothing but a special case of the Minimum Strictly Fundamental Cycle Basis (MSFCB) problem as it has been considered for instance in [DKP82]: Find a spanning tree T that minimizes d T (u, v) + w(e). (3.3) e={u,v} E\T The MAST finds increasing attention in preconditioning, in particular for solving symmetric diagonally dominant linear systems ([EEST05]). There is another related problem for which one can detect an even larger variety in notation. In the Shortest Total Path Length Spanning Tree (STPLST, see [DKP82, WCT00]) problem we seek for a spanning tree that minimizes (u,v) V V d T (u, v). (3.4) The very same problem has also been referred to as the Minimum Routing Cost Spanning Tree (MRCST) problem ([WLB + 99, GA03]). In the special case of an unweighted graph, Johnson et al. ([JLK78]) call it the Simple Network Design problem. Alternatively, when considering complete graphs, Hu ([Hu74]) introduced it as the Optimum Distance Spanning Tree problem. In [DDGS03], the Minimum Average Distance (MAD) spanning tree problem is considered but setting the vertex weights in that model to one, this is another variant of the STPLST problem. Notice that also additive tree spanner problems attracted quite a number of researchers (e.g. [KLM + 03]). However, the work presented in this chapter is restricted to max-stretch and average stretch tree spanner problems.

99 3.2 A Unified Notation for Tree Spanners (UNTS) 93 In the following Section 3.2 we propose a unified notation for the large variety of tree spanner problems. Subject to this notation we identify which problems coincide. More specifically, we consider two problems P and Q to coincide, if every spanning tree T P that is optimum for P constitutes an optimum solution for Q, and vice-versa. We use the notation P Q as a short-hand. Notice that we have to choose this very discriminative equivalence relation. Otherwise, if we allowed for general polynomial transformations, one could no more distinguish between any two NP-complete problems. We provide coincidences for both maximum stretch tree spanners (Section 3.3.1) and average stretch tree spanners (Section 3.4.1), for the cases of graphs with general weights, with metric weights, and with unit weights. We complement our analysis by providing example graphs showing that there are no further coincidences. All the examples consist of fairly small simple 2-vertex connected planar graphs. Consider the very rich world of (in-) approximability results for tree spanner problems, occasionally for special classes of graphs. We expect that having at hand a clear map of the relationships between the various tree spanner problems, a certain cross-fertilization between the different perspectives on much similar structures will occur. In Section we make the first step into this direction. In the context of tree spanners, as recently as 2004 the best known inapproximability factor of the MMST problem has been cited as ([EP04]), being due to [PR99]. During this chapter we show that in the case of unweighted graphs the MMST problem coincides with the Min-Max Strictly Fundamental Cycle Basis (MMSFCB) problem, as it has been stated in [Gal01, Gal03]. There, an inapproximability factor of 2 ε has been achieved already in 2001 ([Gal01]). Hence, this applies immediately to the MMST problem as well. Moreover, in the family of tree spanner problems we identify the only problem whose complexity status has not been identified before. We provide an NP-hardness result for it. 3.2 A Unified Notation for Tree Spanners (UNTS) There are three major criteria in which tree spanner problems may differ: First, either the maximum stretch or the average stretch is to be determined. Second, this objective may be computed with respect to different sets of pairs of vertices, e.g. for (u, v) V V or only for {u, v} E \ T. Third, there have been considered various terms for the objective, e.g. d T (u,v) d G (u,v) or d T(u, v) + w(e). In the remainder, we refer to a tree spanner problem P through a triple (goal, domain, term). We consider the following family of tree spanner problems: goal The goal is either the maximum stretch or the average stretch. domain The domain is either {u, v} E \ T, {u, v} E, or (u, v) V V.

100 94 Classification of Tree Spanner Problems term The term may be one of d T (u, v) + w(e), d T (u, v), d T (u,v) w(e), or d T (u,v) ( ) Notice first that we do not consider, V V, d T (u,v) w(e) d G (u,v). and (, V V, d T (u, v)+ w(e)), because w(e) is not properly defined for (u, v) V V \ E. Second, it could appear somehow strange to count the weight of tree edges twice in the two tree spanner problems (, E, d T (u, v) + w(e)). However, this is consistent with the UNTS. Moreover, this does not cause any degeneracies, because in the next two sections we exhibit that there is always some other tree spanner problem, which coincides with (, E, d T (u, v) + w(e)). Third, observe that for a given graph, E and V V are constant, and E \T is independent of the tree T. Hence, we prefer to represent the goal average with the symbol. We provide a first idea of the wide range of these tree spanner problems by locating several well-known problems of combinatorial optimization within the UNTS: ( max, V V, d T (u,v) d G (u,v) ) is the MMST problem ([CC95]), (max, V V, d T (u, v)) is the Minimum Diameter Spanning Tree (MDST) problem ([HL02]), (, V V, d T (u, v)) is the STPLST problem (or MRCST problem, [JLK78]), and (, E \ T, d T (u, v) + w(e)) is the MSFCB problem ([DKP82]). We will establish that among the remaining 16 problems there is only one single problem which does not coincide with one of these four prominent problems in the case of unweighted graphs. Since its complexity status has not been identified before, we provide an NP-hardness proof for it. However, in the case of weighted graphs there is a much larger variety of problems, in particular in the context of average stretch tree spanners. For instance, in [EEST05] the same techniques are applied to both (, E, d T (u,v) w(e) ) and (, E, d T (u,v) d G (u,v) ). Nevertheless, in general these problems do not coincide. In Figure 3.1 we summarize all the coincidences that exist between tree spanner problems, and which we are going to develop in the remainder of this chapter. Namely, in Sect. 3.3 we deal with max-stretch problems whereas in Section 3.4 average-stretch problems are considered. We organize the sections by subdividing them into three parts where we distinguish general, metric, and unit weights. However, we show that there is a bridge between unweighted and integer-weighted tree spanner problems. Here, we aim at identifying a weighted instance (G, w) immediately with the unweighted instance G that results from replacing every edge e = {u, v} with weight w(e) with a uv-path P e having w(e) edges. Observe that every spanning tree of G has to contain at least w(e) 1 edges of P e. Now, consider the term d T (u, v) + w(e). Let T be some spanning tree of G. We construct a spanning tree T of G such that if e T then P e T. This yields d T (u, v) + w(e) = d T (u, v ) + 1, e = {u, v} E \ T, {u, v } = P e \ T. (3.5)

101 3.3 Maximum Stretch Problems 95 Hence, for the domain E \T in conjunction with the term d T (u, v)+w(e) an optimum solution to a weighted tree spanner problem is obtained by a kind of projection from an optimum solution to the corresponding unweighted problem, and vice-versa. Proposition 3.1. Let goal be a fixed optimization goal. Then, the weighted version and the unweighted version of (goal, E \ T, d T (u, v) + w(e)) coincide. 3.3 Maximum Stretch Problems We start our tour through the zoo of tree spanner problems with maximum stretch tree spanner problems. We first collect the pairs of problems which coincide, where we distinguish between general weights, metric weights, and unit weights. Then, we examine example graphs showing that there are no further coincidences Coincidences It is an elementary observation that if two tree spanner problems coincide even for general weights, in particular they also coincide for metric weights. Moreover, if two problems coincide for metric weights, they immediately coincide for unweighted graphs, too. Hence, to present the coincidences between maximum stretch tree spanner problems, we proceed from the most general weight functions to the most specialized weight function. General Weights. In the case of general weights, there are five families of coincident maximum stretch tree spanner problems. Proposition 3.2. The following two maximum stretch problems coincide: (max, E\ T, d T (u, v) + w(e)) and (max, E, d T (u, v) + w(e)). Proof. Assume for contradiction there was a weighted graph (G, w) such that a spanning tree T that is optimum with respect to (max, E, d T (u, v) + w(e)) attains its maximum exclusively on a tree edge e T. Then, f = {u, v} E \ T : d T (u, v) + w(f) < 2w(e). (3.6) Consider any edge f E \T whose fundamental circuit C contains the edge e. Such an edge exists because G is 2-vertex connected. The total weight of the circuit C is precisely w(c) = d T (u, v)+w(f), and the weight of C \{e} is d T (u, v)+w(f) w(e). By (3.6) there holds d T (u, v) + w(f) w(e) < w(e). (3.7) Now, consider the spanning tree T = T {f}\{e}. Any fundamental circuit (different from C) with respect to T that contained the edge e is replaced with a subpath of C \ {e}. As we only consider positive edge weights, by (3.7) the new fundamental circuit is strictly shorter than the initial one. Hence, the spanning tree T has not been optimum with respect to (max, E, d T (u, v) + w(e)).

102 96 Classification of Tree Spanner Problems d T (u,v) d G (u,v) d T (u,v) d T (u,v)+w(e) Maximum Stretch Tree Spanner E V V E\T unweighted d T (u,v) w(e) E V V E\T Average Stretch Tree Spanner max E\T metric E\T E V V max E V V d T (u,v) d G (u,v) d T (u,v) w(e) d T (u,v) d T (u,v)+w(e) E\T E V V MMST MDST MSFCB max weighted STPLST d T (u,v) d T (u,v) d T (u,v) d T (u,v)+w(e) d G (u,v) w(e) Figure 3.1: A guide to the zoo of tree spanner problems Proposition 3.3. The two problems (max, E \ T, d T (u, v)) and (max, E, d T (u, v)) coincide. Proof. Let T be an arbitrary spanning tree of (G, w). These two problems could

103 3.3 Maximum Stretch Problems 97 only differ, if the maximum in (max, E, d T (u, v)) is attained exclusively by a treeedge e = {u, v} T. But in this case, d T (u, v) = w(e). As we only consider 2-vertex connected graphs, there exists a circuit C through e. The tree T cannot contain all the edges of C. Hence, as we assume the weight function w to be positive, there exists a non-tree edge e = {u, v } C \ T such that d T (u, v) d T (u, v ). ( In the sequel we establish that the following five recall that max, V V, d T (u,v) w(e) ( ) is not properly defined maximum stretch problems coincide: ( max,, d T (u,v) d G (u,v) max,, d T (u,v) w(e) and ). In fact, most of the work was done by Cai and Corneil ([CC95]): Theorem( 3.4 ([CC95]). Consider ) ( the following) five maximum stretch tree spanner problems: max,, d T (u,v) w(e) and max,, d T (u,v) d G (u,v). If for a given weighted graph (G, w) all of them attain an optimum stretch value of t 1, then these five problems coincide on (G, w). However, subject to our definition of coincidence, we are even able to relax the assertion of t being greater or equal than one. To that end, we start with an easy observation. ( ) Lemma 3.5. Consider one of the four tree spanner problems max, E, d T (u,v) w(e) and ( ) max,, d T (u,v) d G (u,v) for some weighted graph (G, w). For the optimum stretch factor t that can be obtained with respect to this problem, there holds t 1. Proof. In the definition of the term d T (u,v) d G (u,v), d G(u, v) is the length of a shortest uvpath in G. Thus d T (u,v) d G (u,v) 1 for all (u, v) V V. When considering theterm d T (u,v) w(e) over thedomain E, for every tree edge e T this edge constitutes the unique uv-path in T. In particular, d T (u,v) w(e) = 1 for all e = {u, v} T E. Proposition 3.6. ( If a weighted graph ) (G, w) admits a tree spanner T such that t < 1 subject to max, E \ T, d T (u,v) w(e), then T is unique optimum for all five problems max,, d T (u,v) ( ) ( ) w(e) and max,, d T (u,v) d G (u,v). Proof. So, let (G,( w) be a weighted graph ) that admits a tree spanner T such that t < 1 subject to max, E \ T, d T (u,v) w(e). Then, in order to prove the proposition it suffices to show that ( ) ( ) 1. for the four problems max, E, d T (u,v) w(e) and max,, d T (u,v) d G (u,v) there holds t = 1; further, ( ) 2. for every spanning tree T T of G the five problems ( ) max,, d T(u,v) w(e) and max,, d T(u,v) d G (u,v) have stretch factor t > 1. )

104 98 Classification of Tree Spanner Problems First, we prove( 1. Therefore notice ) that for each e = {u, v} T it holds d T (u, v) = w(e). Hence, for max, E, d T (u,v) w(e) one immediately observes t = 1. Now, consider ( ) the problem max, V V, d T (u,v) d G (u,v). Let u and v be two vertices of G and let P uv be the unique uv-path in T. Assume for contradiction that P uv is not a shortest uv-path. So, let P be a shortest uv-path in G. Then, P contains at least one edge f = {u, v } that is not contained in T, since otherwise P uv and P contain a cycle in T. However, because of the proposition s assumption we know that d T (u, v ) < w(f). Let P be the u v -path in T. Then this path P can be used to construct an uv-path with length strictly smaller than the length of P. From this contradiction we conclude that for an arbitrary pair of vertices u and v the uv-path in ( T is a shortest uvpath. Hence, d T (u, v) = d G (u, v) and the claim follows for max, V V, d T (u,v) ) d G (u,v) ( ) and thereby for max,, d T (u,v) d G (u,v) with the two remaining domains as well. Now, we prove 2. Therefore, let T and T be defined as in 2. From T T we conclude that there exists some edge e = {u, v} T \ T. In particular, t < 1 provides us with f P uv w(f) < w(e), where P uv T is the unique uv-path in T. Consider the fundamental circuit C T (e) = {e} P uv that the edge e induces with respect to T. As T is a tree, the set of edges F = C T (e) \ T is nonempty, and in particular e F, because e T. Because of w(e) > w(f) for all edges f P uv and since we are only considering positive weight functions w, it remains to detect some edge f F, such that e C T (f). Since the fundamental circuits with respect to T form a basis of the cycle space C(G), and C T (e) C(G), there exists a set F E \ T such that C T (e) = f F C T (f), where we consider the symmetric difference. Due to the special structure of cycle bases that are associated with spanning trees, we know that F C T (e) \ T, in fact F = F ([Ber62]). In particular, as by definition the edge e is contained in C T (e), e has to appear in at least one fundamental circuit C T (f) that is induced by an edge f = {u, v } F. Corollary ( 3.7. ) The following ( five maximum ) stretch tree spanner problems coincide: max,, d T (u,v) w(e) and max,, d T (u,v) d G (u,v). ( Proof. Consider an optimum solution T with respect to max, E \ T, d T (u,v) w(e) ). In the case of a stretch factor t 1 we are done immediately by applying Theorem 3.4. Otherwise, i.e., if t < 1, Proposition 3.6 ensures optimality and uniqueness of T subject to all five optimization problems that we consider here. In particular, all the maximum stretch tree spanner problems that involve fractions coincide.

105 3.3 Maximum Stretch Problems 99 Metric Weights. For maximum stretch tree spanner problems, there are no coincidences in the case of metric weights, which do not apply already to the general case. Unit Weights. For an arbitrary tree T of an unweighted connected graph G (or having weights w = 1) with n vertices and m edges, there holds max {d T(u, v) + w(e)} = max {d T(u, v) + w(e), 2} (3.8) e={u,v} E e={u,v} E\T = max e={u,v} E {d T(u, v) + 1} (3.9) = max e={u,v} E\T {d T(u, v) + 1, 2}. (3.10) Moreover, by w(e) = 1 we obtain immediately d T (u, v) = d T (u,v) w(e). Finally, in the case of an unweighted graph, for every edge e = {u, v} there holds d T (u, v) = d T (u,v) d G (u,v). Together with (3.8) (3.10) and Corollary 3.7 we conclude Proposition 3.8. Let G be an unweighted graph. Except for (max, V V, d T (u, v)), all max-stretch tree spanner problems coincide Anticoincidences In order to prove for two problems that they do not coincide, we profit from the following transitive relation: If the problems do not coincide for unweighted graphs, then they do not coincide for graphs with metric weights. Furthermore, if there is a graph with metric weights for which the sets of optimum solutions for two tree spanner problems have empty intersection, then these problems cannot coincide for general weights either. Thus, we provide the relevant anticoincidences by moving from the most specialized weight function to general weight functions. Unit Weights. As by Proposition 3.8 there are only two different maximum stretch tree spanner problems in the case of unweighted graphs, we only have to establish one single anticoincidence. Example 3.9 (MMST vs. MDST). Consider the unweighted simple graph G in Figure 3.2(a). Recall from Proposition 3.8 and from Theorem 3.4 that in the unweighted case we may think of the MMST problem as (max, E \ T, d T (u, v)). Hence, we are looking for a spanning tree whose non-tree edges are linked by paths in T whose maximum length is minimal. The spanning tree that we highlight in Figure 3.2(b) attains an objective value of two. Moreover, every spanning tree that attains an objective value of two has to induce all five triangles of G as its fundamental circuits. Thus, such a spanning tree must contain the four edges that are not incident with the infinite face. So it must not contain the edge e. In contrast, for that in the MDST problem a diameter of three can be achieved, the leftmost vertex and the rightmost vertex have to be connected via a path of three edges. Observe that there is only one such path. But this includes the edge e, see Figure 3.2(c) for one of the two optimum trees.

106 100 Classification of Tree Spanner Problems e e e (a) (b) Figure 3.2: An unweighted graph and example trees which show that MMST and MDST do not coincide (c) Metric Weights. In order to complement the results of Section 3.3.1, we have to show that the following three problems do not coincide: (max, E \ T, d T (u, v) + w(e)), (max, E \ T, d T (u, v)), and ( ) max, E \ T, d T (u,v) w(e). Fortunately, there exists a fairly small graph with metric weights such that the unique optimal solutions for these three problems are disjoint. Example ( 3.10 ((max, E )\ T, d T (u, v) + w(e)) vs. (max, E \ T, d T (u, v)) vs. max, E \ T, d T (u,v) w(e) ). Consider the graph in Figure 3.3(a). In Table 3.1 the objective values of the three spanning trees in Figures 3.3(b) 3.3(d) with respect to the three objective functions are collected (a) (b) Figure 3.3: A graph with metric weights and its different optima with respect to the objective functions (max, E \ T, goal), where goal {d T (u, v) + w(e), d T (u, v), d T (u,v) w(e) } (c) (d) There are precisely seven circuits in G. One can easily check that the values 10 and 6 are the best values with respect to the objective functions d T (u, v) + w(e) and d T (u, v), respectively, even when considering arbitrary sets of three circuits. Finally, performing a simple inspection of the few relevant cases one can further check that no other spanning tree achieves better values with respect to the three objective functions.

107 3.4 Average Stretch Problems 101 Table 3.1: The values with respect to the different objective functions for the spanning trees in Figures 3.3(b) to 3.3(d) d T (u,v) w(e) Tree d T (u, v) + w(e) d T (u, v) 7 Figure 3.3(b) Figure 3.3(c) Figure 3.3(d) General Weights. As there are no coincidences between maximum stretch tree spanner problems which do only apply to metric weights but not to general weights, this paragraph has to remain void. 3.4 Average Stretch Problems Our tour through the average stretch tree spanner problems follows the trace of our expedition through the maximum stretch tree spanner problems. But we will find many more different problems in the average stretch case Coincidences Comparing the maximum stretch case to the average stretch case on general weights, metric weights, or unit weights, the number of different problems is by up to four larger for average stretch tree spanners. General Weights. There are only two pairs of average stretch tree spanner problems that coincide for general weights. Proposition The two average stretch problems (, E, d T (u, v) + w(e)) and (, E, d T (u, v)) coincide. Proof. For every spanning tree T, the objective values of these two problems differ precisely by e E w(e), being independent of T. Proposition It holds that the average stretch problems (, ) and E, d T (u,v) w(e) coincide. (, ) E \ T, d T (u,v) w(e) Proof. For every spanning tree T, the objective values of these two problems differ precisely by d T (u,v) e T w(e). As for every edge e = {u, v} T the unique path in T between its endpoints is just the edge e, there holds d T (u, v) = w(e). Thus, d T (u,v) e T w(e) = n 1, which again is independent of T. Metric Weights. Much similar to the case of maximum stretch tree spanners, for metric weight functions four problems whose objective functions involve fractions coincide.

108 102 Classification of Tree Spanner Problems Proposition Let (G, w) be an undirected graph with a metric weight function w on the edges. Let domain be either E \T or E, and let term be one of d T (u,v) w(e) and d T (u,v) d G (u,v). Then, the four problems (, domain, term ) coincide. Proof. In the case of a metric weight function w on the edges, for every edge e = {u, v} E there holds d G (u, v) = w(e). Hence, for each of the two domains that we consider here, the two problems (, domain, ) coincide. Moreover, for every tree edge e = {u, v} T there holds d T (u,v) w(e) = 1. Thus, for every spanning tree its objective value with respect to the domain E is precisely n 1 greater than the objective value with respect to the domain E \ T. Unit Weights. With the exception of the average stretch tree spanner problems that are defined for V V, all other average stretch tree spanner problems coincide on unweighted graphs. Similarly to (3.8) (3.10) we find, d T (u, v) + w(e) = d T (u, v) + w(e) + 2(n 1)(3.11) e={u,v} E = = e={u,v} E\T e={u,v} E e={u,v} E\T d T (u, v) + m (3.12) d T (u, v) + m + n 1. (3.13) Again, we profit from the fact that for every edge e = {u, v} there holds d T (u, v) =. Together with (3.11) (3.13) we conclude d T (u,v) w(e) = d T (u,v) d G (u,v) Proposition Let G be an unweighted graph. Then the following eight unweighted tree spanner problems coincide: (, E \ T, ) and (, E, ) Anticoincidences In the case of average stretch tree spanner problems, it will turn out that even for the unweighted case, both problems with domain V V do not coincide with any other problem. Unit Weights. In the case of the most special weights, the following Example 3.15 shows that we remain with 3 problems. (, ) Example 3.15 (MSFCB vs. STPLST vs. V V, d T (u,v) d G (u,v) ). Consider the unweighted planar graph G in Figure 3.4. Observe that the graph from Figure 3.2 can be obtained from G simply by contracting one single edge. Again, the unique minimum cycle basis of G consists of the five circuits which are the boundary of the finite faces of G. Hence, the optimum solution value of the MSFCB problem on G

109 3.4 Average Stretch Problems 103 is 16 and it can be obtained by the fundamental circuits that are induced by eight spanning trees, one of which we display in Figure 3.4(b). These eight spanning trees all contain the four edges of G which are not incident with the infinite face of G, and yield objective values of at least 66 and for the STPLST problem and for (, V V, d T (u,v) d G (u,v) ), respectively. (a) (b) (c) (d) Figure 3.4: An( unweighted graph and example trees which show that none of MSFCB,, ) STPLST, and V V, d T (u,v) d G (u,v) coincide In contrast, the spanning tree in Figure 3.4(c) is one of the four optimum solutions for the STPLST problem. ( Their objective value is 62. On the contrary, for the, ) MSFCB problem and for V V, d T (u,v) d G (u,v) they only achieve objective values of 18 and 228 6, respectively. Finally, Figure 3.4(d) ( shows an example of the four spanning trees which are optimum with respect to V V,, ) d T (u,v) d G (u,v), and which achieve an objective function value of But for the objective functions of MSFCB and STPLST these trees are suboptimal because of objective values of only 17 and 63, respectively. Metric Weights. To discover more anticoincidences we take a look at graphs with a metric weight function. Example 3.16 (MSFCB vs. (, E, d T (u, v)) vs. (, E \ T, d T (u, v))). We investigate the graph G with a metric weight function w that is displayed in Figure 3.5(a). There are precisely two circuits in (G, w) which have weight 18, and another two circuits which have weight 19. There are indeed four spanning trees which achieve an objective function value of = 55 with respect to MSFCB (see e.g. Figure 3.5(b)). But since all of them include two edges of weight seven, they only achieve objective values of 62 and 40 with respect to (, E, d T (u, v)) and (, E\T, d T (u, v)), respectively. In contrast to the optima with respect to MSFCB, there exist two spanning trees which only contain one single edge of weight seven each, but admit the second smallest set of fundamental circuits: = 56. Hence, these are precisely the trees which admit an objective function value of 38, being optimum with respect to (, E \ T, d T (u, v)). One of them is depicted in Figure 3.5(c). Their objective value with respect to (, E, d T (u, v)) is 57. Since we identified all the optima with respect to MSFCB and (, E\T, d T (u, v)), it suffices to provide some spanning tree T that attains a smaller objective function

110 104 Classification of Tree Spanner Problems (a) (b) Figure 3.5: A graph with metric weights and example trees which show that none of MSFCB, (, E \ T, d T (u, v)), and (, E, d T (u, v)) coincide (c) (d) Table 3.2: The objective values to the four considered problems for the trees of Figures 3.6(b) and 3.6(c). Tree (, E \ T, (, E \ T, (, E \ T, (, E, d T (u,v) w(e) ) d T (u, v) + w(e)) d T (u, v)) d T (u, v)) Figure 3.6(b) Figure 3.6(c) value with respect to (, E, d T (u, v)) than the former trees did. Indeed, the spanning tree T that we display in Figure 3.5(d) yields an objective function value of only 56. One can easily observe that T is the unique minimum spanning tree of (G, w). Actually, it is even the unique optimum solution with respect to (, E, d T (u, v)). (, ) Example 3.17 ( E \ T, d T (u,v) vs. {(, E \ T, d T (u, v) + w(e)), (, E \ T, w(e) d T (u, v)), (, E, d T (u, v))}). Consider the graph G of Figure 3.6(a). Because of the very regularly structured weights we need only to consider two families of spanning trees: those that include the edge of weight 9, and those which do not. Within both families then all trees constitute indistinguishable solutions for all the considered problems. Representatives for the families are depicted in Figures 3.6(b) and 3.6(c), respectively. The following table now proves the desired claim: whereas for the fractional problem, (, E \ T, d T (u,v) w(e) ) it does not pay off to include the expensive edge of weight 9, it does for the other three problems. It rather turns out to be good to include this particular edge such that it can be used as a shortcut when considering d T (u, v) for (u, v) = e E \ T. General Weights. At last we need to consider graphs with non-metric weight functions to prove the remaining anticoincidences. (, ) ( Example 3.18 ( E \ T, d T (u,v), ) ( w(e) vs. { E \ T, d T (u,v), ) d G (u,v), E, d T (u,v) d G (u,v) }). Consider the graph with non-metric weights in Figure 3.7. A first observation is that the edge with weight 8 is not metric. More important, the edge with weight 1 is included in every optimal tree for all the three problems. Otherwise we immediately have a contribution of 14 which is the length of a shortest circuit through this

111 3.4 Average Stretch Problems (a) (b) Figure 3.6: A graph with metric weights and example trees which show that none of MSFCB, (, E \ T, d T (u, v)), and ( (, E, d T (u, v)) coincide with, ) E \ T, d T (u,v) w(e) (c) edge whereas any other tree induces shorter circuits w.r.t. both d T (u,v) w(e) even when considering the sum over all edges. So, we remain with 5 different trees. Among these, due to symmetry reasons it suffices to consider only three trees, cf. 3.7(b)-3.7(d). The following table provides the values for the three trees with respect to the different ( problems showing that the tree in 3.7(b) is the unique optimal solution to, ) E \ T, d T (u,v) w(e) whereas the tree indicated in Figure 3.7(c) is optimum for the (, ) ( other two problems, E \ T, d T (u,v), ) d G (u,v) and E, d T (u,v) d G (u,v). d G (u,v) and d T (u,v) Table 3.3: The objective values of the spanning trees in Figure 3.7 w.r.t. the three 1 problems of Example A term of 840 is factored out for clarity reasons. (, ) ( Tree E \ T, d T (u,v), ) ( w(e) E \ T, d T (u,v), ) d G (u,v) E, d T (u,v) d G (u,v) Figure 3.7(b) Figure 3.7(c) Figure 3.7(d) (a) (b) Figure ( 3.7: A weighted graph and example trees. Figures (b) and (c) show, ) ( that E \ T, d T (u,v), ) w(e) does neither coincide with E \ T, d T (u,v) d G (u,v) nor with (, ) E, d T (u,v) d G (u,v). (c) (d)

112 106 Classification of Tree Spanner Problems (, ) Example 3.19 ( E \ T, d T (u,v) vs. d G (u,v) (, ) E, d T (u,v) d G (u,v) ). We will show the anticoincidence of the two tree-spanner problems with the help of the weighted graph (G, w) in Figure 3.8(a). The dots in the figure shall indicate that we assume a sufficiently large number of clips, i.e., 4-circuits that share one common edge e or f, respectively. Let M denote this number. Further, we will refer to the edges with weight equal to 101 as clip-edges. Recall from Proposition 3.13 that the two problems coincide in the case of metric weights. Hence, we chose the weight function w such that precisely one edge is not metric: the edge g. To show the anticoincidence we will argue as follows: in the beginning we show that each spanning tree T that is optimal for any of the two problems must have a certain structure. First, all edges having weight one are included in T, second, T does not contain any clip-edge, and third, the edges e and f are in T. See Figure 3.8(b), where we highlight edges that have to be in T. Edges that are not in T are depicted by dotted lines in this figure. Observe that as we obtain this structure for parts of the graph where all edges are metric, the structural properties apply to the optimum solutions subject to both objective functions that we are investigating in this example. Thereafter, having this common structure, optimal trees to (, E \ T, d T (u,v) d G (u,v) ) and (, E, d T (u,v) d G (u,v) become distinguishable on the remaining part of the graph, where the only nonmetric edge g is going to play a key role. So, we start motivating the mentioned structure of an optimal tree T. We first show that w(a) = 1 implies a T. Notice first that for any of the 2M clips at least one edge of the clip with weight one has to be in T because otherwise the tree T would not be connected. Hence, assume that for a clip exactly one edge of weight one and its clip-edge are contained in T. In that case, however, we get immediately a contradiction to the optimality of T: A simple exchange of the clip-edge by the non-tree edge with weight one within this clip instantly effects a better tree. To see this, observe that for no other pair of vertices in G the corresponding path in T can traverse one of these two edges and compare the according values d T (x,y) d G (x,y). Now, we know w(a) = 1 a T. Next, we establish that w(a) > 100 implies a T in any tree that is optimal with respect to one of the two objectives that we are investigating in this example. To this end, consider one bundle of clips, say the one that contains the edge e, and assume an optimal tree T contains a clip-edge. Since we already know that the edges having weight one are in T, the tree can contain at most one clip-edge, because otherwise the tree T would include a cycle. Similarly, we conclude that e T. Again, such a structure contradicts the optimality of T, because another local change on T improves the tree: This time we exchange the clip-edge c that we assume to be contained in T with the edge e and obtain a different tree T = T {e} \ {c}. This exchange shortens the length of the path in T between the vertices incident to e, thereby shortening the distances of all paths within T that of the non-tree edge e w.r.t. T with the corresponding value of the non-tree edge c w.r.t. T an improvement is obtained. Notice that here we only define the particular tree T to contain the edge e. But so far nothing is said whether e is contained in any optimum contain these two vertices. In addition, even when comparing the value d T (x,y) d G (x,y) )

113 3.4 Average Stretch Problems (a) Figure ( 3.8: A weighted graph on which the optimum solutions for, ) ( E \ T, d T (u,v), ) d G (u,v) and E, d T (u,v) d G (u,v) do not coincide. Whereas for the first objective it( pays off to include the non-metric edge g into an optimal tree an optimal, ) solution to E, d T (u,v) d G (u,v) is attained without the edge g. u g... e (b) f v... tree. The last structural property that we are about to develop for an optimal tree T with respect to any of the two objective functions, is {e, f} T. We already know that T contains all edges of weight one and no clip-edge. Hence, at least one of the edges e and f is in T, because otherwise the tree T was disconnected. So we assume for contradiction without loss of generality that e T but f / T. Then, consider the unique path P between the vertices u and v in T where obviously f T implies e P. Now, an exchange of any of the edges of P by the edge f will lead to a contradiction to the choice of T, which was an optimal tree. One can see this as follows: before the exchange, each f-clip-edge induces a path in T of length d T at least 126. Therefore, in both objectives each f-clip-edges contributes at least M After the exchange i.e., now with f T this amount decreases to M It is clear that we may choose the parameter M so large that this gain compensates the possibly appearing increases of contributions of the remaining part of the graph that is independent of M. This way we force {e, f} T, and in a sense decouple the clip-edges from the remainder of the tree. At this point we developed all of the structural properties of an optimal tree T. Let us emphasize that the properties hold for optima of both objectives, because up to this point we only argued for parts of the graph on which the two objective values differ by a constant term, because the edges within the clips are all metric. For the remainder of the graph we discuss the effect of adding two more edges to our tree T. Observe that there are exactly two spanning trees that contain the edge g and three which do not. We start by computing the objective value K that the three non-tree edges that are distinct from clip-edges contribute. If g T, then K = [14.87, 14.88].

114 108 Classification of Tree Spanner Problems Otherwise, if g T, there are two trees for which K = [14.94, 14.95], and one for which K > 16. Hence, for the domain E \ T, the two trees that contain the expensive non-metric edge g turn out to be optimal. In contrast, for the domain E it does not pay off to include the non-metric edge g into the tree: it costs 10 9 instead of only one for any other tree edge, which is in particular metric, and this extra cost of more than 0.1 gets not compensated by a reduction ( of less than 0.08 in K. Hence,, ) on (G, w) the average stretch tree spanner problem E, d T (u,v) d G (u,v) is optimized by two of the three trees with g T. 3.5 Max-Stretch And Average-Stretch Problems Never Coincide In this section we aim at detecting anticoincidences between max-stretch and averagestretch problems. Therefore we consider unweighted versions of the problems. Example 3.20 (MMST vs. MSFCB). Consider the unweighted simple 2-vertex connected undirected planar graph G in Figure 3.9. We will argue that an optimum solution for the MSFCB problem contains the edge e, whereas an optimum solution for the MMST does not. e e e f f f (a) (b) Figure 3.9: An unweighted graph with representatives of optimal solutions to MMST and MSFCB (c) Consider the MMST problem. We construct a spanning tree T all of whose fundamental circuits have at most four edges, cf. Figure 3.9(b). First, observe that there has to hold f T, because the only 4-circuits through the south-east most and through the south-west most vertices share the edge f. But then, in order to prevent a 5-circuit, the two edges that are incident with f must be contained in T, too. In turn, e T. The five fundamental circuits of such a spanning tree T thus have lengths (3, 4, 4, 4, 4). In contrast, every optimum spanning tree for the MSFCB problem induces fundamental circuits of lengths (3, 3, 3, 4, 5), see Figure 3.9(c) for an example. But this can only be achieved by including the edge e in the spanning tree, because the only three triangles in G all share this edge. Example 3.21 (MDST vs. {STPLST, (, V V, d T d G (u,v))}). Consider the unweighted undirected graph G in Figure We will argue that the set of spanning

115 3.5 Max-Stretch And Average-Stretch Problems Never Coincide 109 trees which are optimum ( for MDST is disjoint from the set of spanning trees optimal, ) for STPLST or V V, d T (u,v) d G (u,v). e u e u e u v v v (a) (b) Figure 3.10: An unweighted graph and parts of( example trees which show that MDST, ) does neither coincide with STPLST nor with V V, d T (u,v) d G (u,v) (c) We start by detecting a necessary condition for a spanning tree to be optimum for MDST. To that end, first observe that an optimum spanning tree T with respect to MDST achieves a diameter of four. Consider the vertex u. The unique shortest circuit C through u has five edges, whereas the second-shortest circuit through u has six edges. Hence, in a minimum diameter spanning tree T in G, the circuit C is the only fundamental circuit with respect to T that contains the vertex u. But since G\{u} is 2-vertex connected, this implies the three bold edges in Figure 3.10(b) to be contained in T, in particular e T. In contrast, one can check that each of the twelve spanning trees which are optimum for STPLST, with objective value 86, is also optimum for (, V V, d T (u,v) d G (u,v) having objective value 54, and vice-versa. Moreover, in each of these trees there holds δ({v}) T, where δ({v}) is the cut induced by v. In particular, e T. Finally, the next example covers the remaining anticoincidences. ), ) }). (, Example 3.22 ({MMST, MSFCB} vs. {MDST, STPLST, V V, d T (u,v) d G (u,v) A key( difference between MMST and MSFCB on the one side, and MDST, STPLST,, ) and V V, d T (u,v) d G (u,v) on the other side, is that the former problems may be regarded as to have only the set of non-tree edges E \ T as domain, whereas the latter have V V as domain. But only the latter ensures a kind of global perspective for every spanning tree. With E \T as domain, an accordion-like tree as the one that we already displayed in Figure 3.2(b) admits a much more local way of counting. Consider again the unweighted graph in Figure 3.2. By the very same arguments which showed that e T for every spanning tree T which is optimum for MMST, this edge is also contained in every spanning tree which is optimum for MSFCB. More precisely, the four spanning trees which are optimum for MMST are precisely the optimum solutions for MSFCB these four spanning trees only differ in how the left- and the rightmost vertex is connected to the tree.

116 110 Classification of Tree Spanner Problems Similarly, the two spanning trees of minimum diameter are precisely ( the optimum, ) solutions for STPLST, having objective value 42 and even for V V, d T (u,v) d G (u,v), having objective value 28. These two trees only differ in which endpoint of the edge e becomes the vertex of degree four in T. To summarize this section, there is not one single bridge between Max-Stretch and Average Stretch tree spanner problems. In contrast, there exists a related pair of problems for which such a bridge between the maximum objective and the sum objective was established. In [CGH95] it has been shown that any solution to the general Minimum Cycle Basis (MCB) problem is also a solution to the problem of computing a cycle basis whose longest circuit is minimum. Here, it is well-known that the MCB problem can be solved in polynomial time ([Hor87]), more precisely in O(m 2 n + mn 2 log n) ([KMMP04]). 3.6 First Benefit of the UNTS Recall that whenever a spanner problem (goal, domain, term) is NP-hard in the unweighted case, it is in particular NP-hard in its weighted versions, too. We are aware of three such negative complexity results for unweighted tree spanner problems. Theorem 3.23 ([JLK78]). The STPLST problem is NP-hard. Theorem 3.24 ([DKP82]). The MSFCB problem is NP-hard. Theorem 3.25 ([CC95]). The MMST problem is NP-hard. There have even been identified special classes of graphs on which these problems are still NP-hard. For instance, think of the MMST problem on planar graphs (see [FK01]), on chordal graphs ([BDLL04]), and on chordal bipartite graphs (see [BDL + 03]). But there is also one positive complexity result that has been obtained for a tree spanner problem. Theorem 3.26 ([HT95]). The weighted MDST problem can be solved in O(mn + n 2 log n) time. Now, the UNTS provides us with a clear perspective on 20 tree spanner problems. In particular, Examples 3.15, 3.22, ( and 3.21 establish that none of the above, ) complexity results apply to the problem V V, d T (u,v) d G (u,v). Fortunately, we are able to settle its complexity status in Section by establishing NP-hardness. Moreover, by Proposition 3.8 we know that in the ( case of unweighted) graphs the classical maximum stretch tree spanner problem max, V V, d T(u,v) d G (u,v) coincides with (max, E \ T, d T (u, v) + w(e)). In Section we compare two inapproximability results that have been obtained for these problems. Interestingly, the result which had never been stated before in the language of tree spanners turns out to be stronger.

117 3.6 First Benefit of the UNTS An Open Complexity Status Theorem The average stretch tree spanner problem NP-hard. (, ) V V, d T (u,v) d G (u,v) is Proof. As our proof is similar to the proof for Theorem 3.23 as it has been given by Johnson et al. ([JLK78]), we adopt the notation of [JLK78]. We consider the following problem, which is known to be NP-hard (Problem SP2 in [GJ79]): Exact Cover By 3-Sets (X3C): Given a family S = (σ 1,...,σ s ) of 3-element subsets of a set T = (τ 1,...,τ 3t ). Does there exist a subfamily S S of sets with pairwise empty intersection, such that σ S σ = T? Given an instance I of X3C, we define an unweighted simple undirected graph G = (V, E) as follows, see Figure 3.11 for an example: R = {ρ 0, ρ 1,...,ρ r }, where r := T S T + 1, R 0 = {ρ 0 }, R = R \ R 0, V = R S T, E = {{ρ i, ρ 0 } : i = 1,...,r} {{ρ 0, σ} : σ S} {{σ, τ} : τ σ S}, ρ 1,...,ρ r... ρ 0 S T (, ) Figure 3.11: The instance of V V, d T (u,v) d G (u,v) that we associate with the instance {(1, 2, 3), (3, 4, 5), (4, 5, 7), (6, 8, 9), (7, 8, 9)} of X3C Denote by X the number of pairs of elements of T that are not contained together in any of the sets of S, i.e., X := {(τ 1, τ 2 ) V V : σ S : τ 1 σ or τ 2 σ}. (3.14) (, ) We will prove that V V, d T (u,v) d G (u,v) has a solution of value at most V 2 V + T S 5 T X, (3.15) if and only if the answer to the instance I is YES. We denote a spanning tree of G that contains the edge {ρ 0, σ} for all σ S a star tree.

118 112 Classification of Tree Spanner Problems Claim. Every optimum solution of (, ) V V, d T (u,v) d G (u,v) is a star tree. Claim. In Table 3.4, we investigate the distances between any pair of nodes in G, in an arbitrary star tree F, and in an arbitrary non-star tree F. In the case of a non-star tree F, there exists one vertex s S such that d F (ρ i, s ) 4 for all i = 1,...,r. In particular, d F (ρ i,s ) d G (ρ i,s ) 2, whereas d F (ρ i,s ) d G (ρ i,s ) = 1, cf. the ( )- entry in Table 3.4. Hence, when considering R S, by the choice of r the objective value of F is by at least T S T + 1 larger than the one of F. Table 3.4: Distances between pairs (u, v) of nodes within the graph. The first column denotes the cardinality of the considered subset of V V. set of u set of v number of node pairs d G (u, v) d F (u, v) d F (u, v) R 0 R T S T R 0 S S R 0 T T R R ( T S T + 1) ( T S T ) R S ( T S T + 1) S 2 2 ( ) R T ( T S T + 1) T S S S ( S 1) S T S T {1, 3} 3 1 T T T ( T 1) X T T X According to Table 3.4 this can only be compensated on S T and on T T. But there, for (u, v) S T there holds and for (u, v) T T there holds d F (u, v) d G (u, v) d F(u, v) d G (u, v) + 2, d F (u, v) d G (u, v) d F(u, v) d G (u, v) + 1. Hence, any non-star tree F can only gain T S T on S T and T T, which is strictly smaller than its loss on R S. (, ) Now that we know that an optimum solution to V V, d T (u,v) d G (u,v) is always a star tree, we will compute the objective value of an arbitrary star tree F. According to Table 3.4, it remains to investigate in detail pairs of vertices from the sets S T

119 3.6 First Benefit of the UNTS 113 and T T. We first examine the set S T and compute for an arbitrary star tree F (u,v) S T d F (u, v) d G (u, v) = (u,v) S T d F (u,v)=1 d F (u, v) d G (u, v) + (u,v) S T d G (u,v)=1, d F (u,v)=3 = T + S ( T 3) + (u,v) S T d G (u,v)=3 d F (u, v) d G (u, v) d F (u, v) d G (u, v) + (u,v) S T d G (u,v)=1, d F (u,v)=3 = T + S ( T 3) + 3 (3 S T )) = S T 2 T + 6 S. d F (u, v) d G (u, v) As this value is independent of F, we conclude that any two star trees differ in their objective value only for pairs (u, v) T T. Recall the definition of X in (3.14). Among the pairs (u, v) T T, there are precisely X for which d G (u, v) = 4 and thus d F (u, v) = 4 and precisely T 2 T X for which d G (u, v) = 2. As F is a star tree, we know that d F (u, v) {2, 4} for every (u, v) T T. Recall that the quantity X only depends on the instance I of X3C. Hence, a spanning tree F is optimum for (, V V, d T (u,v) d G (u,v) if and only if it is a star tree that maximizes the number Y of pairs (u, v) T T for which d F (u, v) = 2. How large can Y get? We prefer to account for the quantity Y from an alternative perspective. To that end, consider the edges FST := F (S T). Note that FST = T, because F is a star tree. Now, we define a function p(e) for the edges e = (σ, τ) FST, 2, if δ F (σ) = 4, p(e) = 1, if δ F (σ) = 3, and 0, otherwise. Thereby, Y = e FST p(e). Then the following statements are equivalent: Y = 2 T. For all e FST, p(e) = 2. For all σ S, δ F (σ) {1, 4}. Finally, we provide a bijection between star trees F of G with δ F (σ) {1, 4}, for all σ S, and Exact 3-Covers S as follows: S (F ) := {σ S : δ F (σ) = 4} and F ST(S ) := S T. A direct computation reveals that the total objective value of the optimum solution for a graph corresponding to a YES-instances of X3C is right as given in Equation (3.15). ),

120 114 Classification of Tree Spanner Problems Inapproximability of the MMST Problem Peleg and Reshef (1999, [PR99]) prove that the MMST problem cannot be approximated within a factor better than (1 + 5)/2, unless P = N P. Even recently, this result is usually cited when illustrating the complexity of the MMST problem ([EEST05]). In Section 3.3, using the UNTS to classify the large variety of similar tree spanner problems, we were able to establish that in the case of unweighted graphs the following four problems coincide: ( the MMST problem ( the problem max, V V, d T (u,v) d G (u,v) max, E \ T, d T (u,v) d G (u,v) ), the problem (max, E \ T, d T (u, v)), and the problem (max, E \ T, d T (u, v) + w(e)). It is a simple observation that in the case of unweighted graphs, for every fixed tree the objective values of the first three tree spanner problems coincide and differ by one from the fourth one. In particular, any constant inapproximability factor that is obtained for one of these problems carries over to the other problems. Now, Galbiati (2001, 2003 [Gal01, Gal03]) investigated the problem (max, E \ T, d T (u, v)+w(e)), which she denotes the Min-Max Strictly Fundamental Cycle Basis (MMSFCB) problem. This culminates in the following theorem for unweighted graphs. Theorem 3.28 ([Gal01, Gal03]). The problem of finding in a uniform graph G a spanning tree that is optimal for (max, E \ T, d T (u, v) + w(e)) cannot be approximated within 2 ǫ, ǫ > 0, unless P = NP. The above considerations enable us to identify the constant inapproximability factor of Theorem 3.28 as a stronger inapproximability ( factor for the Minimum ) Max- Stretch Spanning Tree (MMST) problem, or max, V V, d T (u,v) d G (u,v) in UNTS. Corollary The Minimum Max-Stretch Spanning Tree problem cannot be approximated within a factor better than 2, unless P = NP. ), 3.7 Conclusions and Open Questions In this chapter we presented a unified notation for tree spanner (UNTS) problems. This allowed us to detect that several tree spanner problems coincide. This is complemented by a number of example graphs showing that no further coincidences exist.

121 3.7 Conclusions and Open Questions 115 We even( identified a tree spanner problem, whose complexity status has been open, ) before: V V, d T (u,v) d G (u,v). For this problem, we presented an NP-hardness proof. Moreover, the UNTS enabled us to build the bridge between the cycle bases perspective and the tree spanner perspective on the very same problems. In particular, we establish that the inapproximability result due to Galbiati ([Gal01, Gal03]) initially obtained for the Min-Max Strictly Fundamental Cycle Basis (MMS- FCB) problem applies to the Minimum Max-Stretch Spanning Tree (MMST) problem, too, and outperforms the best inapproximability result that was known in this context: 2 ε compared to However, there is still some work to do in order to draw the complete map of (in- ) approximability results for tree spanner problems, even more when considering more graph classes. Nevertheless, when exploring the wide area of discrete mathematics, we hope the UNTS to provide an accurate common language in order to prevent double work.

122

123 4 EXPERIMENTS With this chapter we come full circle back to Chapter 1. We conduct different computational experiments that all together help to evaluate our NSC model of Chapter 1 and its computational behavior, respectively. More precisely, we run the following three series of experiments: first, in Section 4.1, we compare the computation performances of the three MIP solver CPLEX [CPL07], MOPS [MOP07] and SCIP [Ach07] on selected NSC instances. Second, Section 4.2 is devoted to experiments that evaluate the influence of cycle bases on the computational behavior of the according NSC MIP. Finally, in Section 4.3 we evaluate the NSC model that we developed in Section 1.3. Namely, we use microsimulation in order to judge the quality of the model s solution as well as to compare the solution, i.e., the set of offsets, to existing coordinations and to solutions obtained by other means. 4.1 A MIP Solver Comparison on Selected NSC Instances The experiments in this section pursue two equally important objectives. First, by comparing the running times of the MIPs of different sized example networks we aim to detect the crucial size of a network, i.e., the size up to which good primal solutions can still be found in reasonable time. Both of the terms good and reasonable time will be specified later on. The second objective is to compare MIP solvers. This emerges as an important task when it comes to the practical application of the NSC models, see Section 1.4. However, a MIP solver comparison is of interest by itself, too. Within the experiments for comparing MIP solver we focus on one particular aspect: the quality of upper bounds during the MIP computation. That is to say, we evaluate the quality only of primal solutions. The quality of the lower bounds during computation or even optimality issues are not considered. The reason why we do so is that in applications, such as described for example in Section 1.4, good solutions are of primary importance; lower bounds are of secondary value. For example, when 117

124 118 Experiments using the NSC MIP to optimize the coordination within the iterative optimization scheme, as proposed in Section 1.4, traffic engineers consider solutions still to be good that are around 4 7% off the optimum. Moreover, computation times on the time scale of seconds are considered reasonable. Besides the state-of-the-art MIP solver CPLEX [CPL07], we included the commercial solver MOPS [MOP07] as well as the solver non-commercial SCIP [Ach07] in our experiments. Fortunately, we were able to perform the tests with the latest versions of all three solvers. Namely, we used CPLEX version 10.1, MOPS with version 8.19 and SCIP 1.0. Whereas CPLEX and MOPS come with an integrated LP solver, we used SoPlex [Wun96] in version as LP solver within SCIP. We chose SoPlex, because we wanted to consider one non-commercial MIP and LP solver package. The procedure of the experiments is as follows: first, we choose example instances, a discussion of which is located below. Then, for each of the instances we run all three solvers with different time limits. In fact, we conduct two series of calculations: First, we ran the MIP solvers with their default settings and, second, we used special solver parameter calibrations that benefited the finding of good solutions. In detail, these special solver parameters were: in CPLEX: mipemphasis = 4 in MOPS: a composition 1 of parameters regulating cuts and heuristics [Suh07] in SCIP: a composition 1 of parameters regulating cuts and heuristics [AB07]. Noticeably, however, there is no one single parameter set that benefits primal computation behavior. For example, the CPLEX parameter mipemphasis = 4 is actually meant for finding hidden feasible solutions. Still, it brought solutions with the best values. For tuning the solver parameters of MOPS and SCIP we followed suggestions by Suhl [Suh07] and Achterberg and Berthold [AB07], respectively. In both cases parameters were adjusted on the basis of our NSC instances. Furthermore, we were not able to use uniform computer architectures on which to run the experiments. In particular, CPLEX and SCIP run on an Intel P4 machine with 3.2GHz, 1.0GB RAM running Linux. On the other hand, MOPS runs under Windows XP on an Intel P-M 2.13GHz with 1.0GB RAM. This inconvenience is due to different license requirements of the three solvers and should be kept in mind when valuating the results in Tables 4.2 to 4.3. As test instances, we chose six specially tailored subnetworks of the Denver network, which is real-world data provided to us by the PTV [ptv]. See Section for more information on data acquisition. The instances were built such that the whole spectrum of network sizes is covered: we consider a small network, medium size networks and large networks. Although all six networks originated from the same instance, the vehicle load within the networks and the networks topologies differ. Hence, the instances can be considered to be of sufficiently distinct characteristics. Information on the particular networks and the according mixed-integer programs 1 At the end of the chapter we quote the explicit list of parameters.

125 4.1 A MIP Solver Comparison on Selected NSC Instances 119 can be found in Table 4.1. For general facts about the Denver network and the other real-world instances we refer to Section 4.3. Table 4.1: A list of parameters, MIP and network, respectively, for the selected instances. The abbreviation Vars hereby refers to the number of variables in the MIP. The acronyms Box, Int, Bin stand for the different types of variables: namely, boxed variables, integer variables and binary variables. Moreover, the number of constraints and the number of non-zero elements, NZs, are quoted. In addition, the number of nodes, the number of edges, and the dimension of the cycle space of the graph, ν, are given, as the most important network parameters. Param Denver Small Medium 1 Medium 2 Medium 3 Big Complete #Vars ,500 2,031 #Box ,086 #Int #Bin #Constraints ,128 1,759 2,403 #NZs 412 1,460 1,708 2,572 4,148 5,783 #Nodes #Edges ν The Tables 4.2 to 4.3 show the results of the experiments. The numbers displayed there are to be read as follows: for each instance, solver, and time limit we state two quantities. First, we give the objective value, that is, the best feasible solution up to that time limit. Since absolute values are hard to judge, we provide for comparison the relative gap to the optimal value or, if the optimum is not known, the relative gap to the best known lower bounds. Information on the respectively used lower bound are stated behind the name of the respective instance. So, notice that our gap does not coincide with the usual gap during calculations. By usual gap we mean the following: instead of using the formula 100 (UB t LB t )/UB t 2 to calculate the gap at time t, where UB t, and LB t denote the best upper and lower bound at time t, we rather use a global lower bound LB that is independent of t for the gap calculation: 100 (UB t LB)/UB t. For all other instances with the exception of Denver Big and Denver Complete we took the optimal value as LB. For the instances Denver Big and Denver Complete, we set LB to the value of the lower bound obtained by a CPLEX run of 3 hours with an emphasis on moving best bound. We decided to introduce this special gap since it better evaluates the primal solutions, which we are exclusively interested in. In the remainder of this section we will analyze the results depicted in Table 4.2, 2 Notice that this is only true for CPLEX and MOPS. The solver SCIP uses the formula 100 (UB t LB t ) / LB t instead.

126 120 Experiments Table 4.2: A comparison of solutions of example instances by selected MIP solvers. The asterisk at the lower bound values marks an optimal solution. For each solver, the only changed solver parameter was the timelimit parameter. All other parameter were left to their DEFAULT values. A indicates, that no feasible solution was found w.r.t. the particular time limit. Moreover the # means that optimality was proven. See also Remark 4.1. Time CPLEX MOPS SCIP limit Denver Small LB = s # % # % # % Denver Medium 1 LB = s % % 5s % % % 10s % % % 30s % % % 60s # % % % 600s % % Denver Medium 2 LB = s % % 5s % % % 10s % % % 30s % % % 60s % % % 600s % % % Denver Medium 3 LB = s % % 5s % % 10s % % 30s % % % 60s % % % 600s % % % Denver Big LB = s % 5s % % 10s % % 30s % % 60s % % 600s % % Denver Complete LB = s % 5s % 10s % 30s % 60s % 600s %

127 4.1 A MIP Solver Comparison on Selected NSC Instances 121 Table 4.3: A comparison of solutions of example instances by selected MIP solvers. The asterisk at the lower bound values marks an optimal solution. For each solver a TUNED parameter set was used. A indicates, that no feasible solution was found w.r.t. the particular time limit. Moreover the # means that optimality was proven. See also Remark 4.1. Time CPLEX MOPS SCIP limit Denver Small LB = s # % # % # % Denver Medium 1 LB = s % % % 5s % % % 10s % % % 30s % % % 60s % % % 600s % % % Denver Medium 2 LB = s % % % 5s % % % 10s % % % 30s % % % 60s % % % 600s % % % Denver Medium 3 LB = s % % % 5s % % % 10s % % % 30s % % % 60s % % % 600s % % % Denver Big LB = s % 5s % % % 10s % % % 30s % % % 60s % % % 600s % % % Denver Complete LB = s % 5s % 10s % % 30s % % 60s % % % 600s % % %

128 122 Experiments results with solvers running with default settings, and in Table 4.3, results with solvers running with tuned settings. It is not at all surprising that in all instances and for all time limits CPLEX outperforms MOPS and SCIP. Especially with default solver settings, CPLEX outpaces the other two solvers that for the larger instances have severe difficulties to find a feasible solution at all. When comparing MOPS and SCIP with their default settings, the following attracts attention. First, their performance is quite comparable in the first, i.e., smallest, three instances. For the larger instances, however, SCIP fails to provide feasible solutions. Here, MOPS performs a little better, that is to say, comes up with feasible solutions. Nevertheless, they are mostly of poor quality. In this context, the following is remarkable: we defined the NSC problem, see Section 1.2.1, using node offsets. So, when modeling the NSC with node offsets, finding a feasible solution should be unproblematic, since any solution is feasible. However, because we decided to use link offsets for our NSC MIP (1.25), this is no longer the case. Nevertheless, although not explicitly stated in our model, the Equations (1.2) that couple the node offsets and the link offsets are inherent in the NSC MIP (1.25). They propose a trivial way to find a feasible solution: for example, the all zero node offset vector can be easily transformed into the link offset vector. Hence, the fact that MOPS and SCIP for the larger instances fail to come up fast with a feasible solution, shows that these solvers cannot detect the implicit connection (1.2). Now, let us remember that for the two largest instances, Denver Big and Denver Complete, we do not know the value of an optimal solution. Therefore, we had to use a rather poor lower bound. Nevertheless, we think that the gaps in fact are misleading, and the obtained solutions are closer to the optimum value than the percentages reveal. When considering Table 4.3 with the results for the runs with tuned parameter settings one notices the following. All three solvers perform better compared to the runs with their default settings. The difference becomes apparent especially for MOPS and SCIP. With the tuned parameter settings feasible solutions are found for all instances, albeit not as fast as CPLEX. In the beginning of this section, we raised the question of the crucial size of a network. When analyzing the results of the experiments we observe that providing good solutions in reasonable time for medium size instances is not problematic for the state of the art MIP solver CPLEX. In fact, we believe that even for Denver Big and Denver Complete the values in Tab. 4.3 of solutions obtained by CPLEX are of satisfying quality, too. Thus, we conclude that finding good solutions for real world NSC instances of sizes up to approx. 150 signals is possible. As for the other solvers, MOPS and SCIP, the results show that small to medium size networks are tractable. Nevertheless, with a more evolved solver parameter tuning both MOPS and SCIP may perform better. During the experiments both solvers, but especially SCIP, showed a remarkable sensitivity towards the adjustment of the solver parameter settings. Notice, though, that discussion of a crucial size of a network only refers to the application of the model within the optimization scheme proposed

129 4.2 The Influence of Cycle Bases on the MIP Performance 123 in Sect When using the optimization as a stand-alone a longer computation time is acceptable and, thus, larger instances are tractable. In summary: among the three MIP solver CPLEX, MOPS and SCIP, the solver CPLEX performs best on the selected NSC instances, whereas MOPS and SCIP perform on a similar level. Remember, however, that the experiments in this section do not purport to be a general solver comparison. Rather, they provide an example of the solvers performances on special instances, i.e., instances from an NSC model, with a clear focus on the quality of primal solutions. We conclude the section with the following remark. Remark 4.1. For all the experiments, whose results are listed in Tables 4.2 to 4.3, we used the time limit parameter of the solvers to limit the computation time of the solver. However, in the case of CPLEX, setting this parameter to a particular value changes the computational behavior. That is, a certain fraction of the available time is saved in order to scan the branch and bound tree for good primal solutions at the end of the computation. However, this might result in a non-intuitive nonmonotonicity of the progression of solutions. That is to say, that more time does not necessarily bring equally good solutions, since the final heuristic can not be anticipated. See, for example, the results for the instance Denver Medium 2 in Tab In the solvers SCIP and MOPS the time limit parameter does not have such side effects. 4.2 The Influence of Cycle Bases on the MIP Performance In this section we conduct experiments that explore the influence of cycle bases on the performance of a mixed-integer linear program. That is to say, we build for selected example networks the according NSC formulation using different types of cycle bases and evaluate the MIP performance. First we discuss the reasoning behind this kind of experiment and the experimental settings, i.e., the chosen criteria for measuring the quality of a cycle basis and measuring the MIP performance, the selected instances and cycle bases. For many real-world applications cycle bases are needed to build appropriate models. That means, often when modeling problems via a graph, a structural property has to hold for all cycles in a graph. Obviously, one does not want to formulate constraints for all cycles. Then it is equivalent, to formulate the needed constraints for a subset of the cycles: a cycle basis. It can be shown that, for several applications, different cycle bases are in use to maintain a concise problem formulation, see Chapter 1 and [Gle01, Lie06, Bol02]. However, since there is a huge number of different cycle bases, one question naturally arises: Are all cycle bases equally good, or do some bases result in a better problem formulation? The possible reasoning behind this question is the following: when we look at the problem formulation, often cycle bases are used to model some kind of periodicity. Then, as it can also be observed in our NSC MIP (1.25), those periodicity constraints demand integer variables for which one wants to give

130 124 Experiments good bounds; naturally, good bounds immediately reduce the search space and induce a correspondingly tighter problem formulation. For example, Equations (1.9) and (1.10) show how bounds for the NSC model can be obtained. Here, of course, different cycle bases yield different bounds on the integer variables. The hope is that cycle bases that allow for tight bounds, and thus result in a tighter problem formulation, lead to MIPs that perform better. Although the above reasoning is neither surprising nor new, there are hardly no studies about the real influence of bases on the performance of the according MIP. Only Liebchen [Lie06] reports on that topic in the context of modeling periodic timetables. We chose as test instances three subnetworks of the Denver network with different sizes, namely the instances Denver Medium 3, Denver Big and Denver Complete. For data on the networks and the according mixed-integer linear programs see Table 4.1. Notice, that the data given there does not depend on the selected cycle basis. A canonical quality measure of a cycle basis when trying to anticipate the computational behavior of its MIP is the width W(B) of a cycle basis B, see [Lie06]. It is defined as W(B) := l B ( nl n l + 1 ). (4.1) Hereby denote n l and n l an upper and lower bound, respectively, on the integer variable n l of the NSC MIP constraint of the cycle l B. See Equations (1.9) and (1.10) in Section However, just as Liebchen suggests in [Lie06], we use W (B) := log W(B) = l B ( log(nl n l + 1) ) (4.2) since this way we avoid the uncomfortable product in (4.1). Notice that replacing W(.) by W (.) is valid, because the log-function is monotonically increasing. The importance of the width of a cycle basis is nicely summarized by Liebchen who concludes that the width of a cycle basis suggests how CPLEX might perform on the resulting integer program [Lie06]. One aim of the experiments in this section is to investigate to what extent this observation can be shared. It is worth mentioning, however, that other quality measures of a cycle basis for example its length, w.r.t. a certain cost function or w.r.t. the total number of edges are possible, too. We decided for the width of a basis as measure because it has the most immediate connection to the MIP, as it predicts its search space. Another crucial question is how the influence of a cycle basis on the performance of the according MIP can actually be determined. That is to say, to which MIP measure we do relate the width W(B) or W (B), respectively, of the considered basis B? Among possibilities like total computation time or value of the according LP relaxation we decided to take the lower bound after 10 seconds as criteria. We did this for the following reason. The value of the LP relaxation has the major shortcoming of having a very narrow range of values. Namely, we observed a range of approximately 2.5% within all occurring LP relaxation values for experiments on

131 4.2 The Influence of Cycle Bases on the MIP Performance 125 the three considered instances. On the other hand, considering the total computation time was not possible for the selected instances. In addition, we feel that considering the lower bound of the MIP (after a short time of computation) better reflects the pure connection between the cycle basis and its MIP since it blanks out primal heuristics. The choice of the time limit of 10 seconds then was also a concession to the great number of single experiments that we were about to conduct. Other values for the time limit are possible, however. In the next paragraph we specify which cycle bases we considered for our experiments. We chose two specific single bases and two series of bases. Namely we examine: 1. a strictly fundamental cycle basis that is induced by a Minimum Spanning Tree of the graph with the negative of the edge multiplicities as cost function on the edges. In Fig. 4.2 this tree is denoted by B1. 2. A second single basis denoted by B2 that is built from B1 by the following heuristic: Consider the spanning tree T that induces B1. Further, sequently consider branches e / T. If e induces a cycle of length 2 or has no parallel edges do nothing. Otherwise, substitute for each edge f / T parallel to e its induced cycle of length 3 by the length 2 cycle consisting of the edges e and f. See Fig. 4.1 for an example. It is obvious that then B2 is a basis. Moreover it can be shown that B2 is weakly fundamental and hereby integral. 3. Furthermore we consider a series of 500 bases induced by random spanning trees. We constructed a random spanning tree by putting random integer costs (between 1 and 10 3 ) on the edges and computing a minimum spanning tree. However, for parallel edges we chose the same random cost. We do so, because otherwise edges with high multiplicity are likely to be in the tree and, hence, the induced basis is too similar to the one described in 1. In Fig. 4.2 these trees are collected under rand-st. 4. Finally, we take the very same 500 bases as in 3 as starting bases and refine them individually with the heuristic described in 2. In Fig. 4.2 these trees are denoted by WFCB-heuristic. Notice that the bases described in 1 and 3 are strictly fundamental whereas those from 2 and 4 are only weakly fundamental. All bases considered are integral. We conclude the description of the experimental settings by mentioning that the experiments were performed on an Intel P4 machine with 3.2GHz, 1.0GB RAM running Linux. As MIP solver we used CPLEX in version 10.1 [CPL07]. In Figures 4.2(a) to 4.2(c) the results of the experiments are collected and they will be discussed in the remainder of the section. As can be seen from the diagrams 4.2(a) to 4.2(c) the obtained results are consistent with respect to the different instances. In all three cases we observe that the random spanning trees induce bases with a much greater width than their weakly fundamental counterparts; this, however, is not surprising. Neither it is surprising that basis B1 has a far smaller width than the average strictly fundamental one.

132 126 Experiments e 5 e 6 e 7 e 8 e 4 e 3 (a) G e 2 e 1 e 5 e 7 (b) T e 1 e 7 e 8 e e 5 e 7 6 e 2 e 1 e 1 e 5 e 4 (c) B \ {e 5, e 7, e 1, e 3 } e 5 e 6 e 7 e 8 e 4 e 3 e 1 e 2 (d) B \ {e 5, e 7, e 1, e 3 } Figure 4.1: An illustration of the WFCB heuristic is shown. Consider the graph G, (a), a spanning tree T of G, (b), and the strictly fundamental cycle basis induced by T, (c), where we omitted the cycle {e 5, e 7, e 1, e 3 } for clarity reasons. Then B, (d), is a result of the WFCB heuristic. In particular, only one cycle was exchanged, {e 5, e 7, e 1, e 4 } for {e 3, e 4 }. Notice that B is no longer strictly fundamental. After all, B1 was built such that edges with high multiplicity are likely to be in the tree, which, in turn, results in many short, i.e., length 2, cycles. In addition, the following fact stresses this observation: when considering a minimum spanning tree with positive multiplicities as costs on the edges one gets the following three tuples of (W (B), lower bound after 10 seconds of MIP computing ) for the three instances: (291.26, ), (471.56, ) and (657.65, ). The numbers show that these bases have a huge width and, not forestalling too much, inferior lower bounds compared to the considered bases. For clarity reasons, we omitted theses bad examples in Fig In considering the results plotted in Figs. 4.2(a) to 4.2(c) our conclusion is twofold. First, and unfortunately, a clear straight correlation of the width of a basis to the MIP performance, i.e., the MIP lower bound after 10 seconds of computation, can not be affirmed. Namely, a basis with a smaller width does not necessarily lead to a MIP with better, i.e., higher lower bound. Notice that this would have meant that the pairs of variates string along a monotonically decreasing function. However, this can not be observed in the diagrams. For example, for all three instances there are pairs of bases whose widths are almost equal but whose resulting lower bounds diverge significantly. See for example in 4.2(a) bases with tuples (251.59, ) and (254.09, ). Still, we draw following conclusions: when comparing the series of weakly fundamental bases to the series of random strictly fundamental ones, the weakly fundamental bases perform better. Namely, taking the average on the lower bounds of the weakly fundamental bases, it is by 8.0%, 5.9%, and 7.4% (for the three instances) higher than the corresponding quantity of the strictly fundamental bases. The same holds when investigating the median. The median of the lower bounds of the weakly fundamental bases is by 8.4%, 5.5%, and 7.3% higher than the median of the lower bounds of the strictly fundamental bases. Thus, one could interpret this as a strong indication that improving the strictly fundamental bases, even though they become weakly fundamental, and thereby reaching a whole smaller level

133 4.2 The Influence of Cycle Bases on the MIP Performance rand-st WFCB-heuristic B1 B MIP lower bound after 10 seconds log of width of cycle basis (a) Denver Medium rand-st WFCB-heuristic B1 B2 MIP lower bound after 10 seconds log of width of cycle basis (b) Denver Big rand-st WFCB-heuristic B1 B2 MIP lower bound after 10 seconds log of width of cycle basis (c) Denver Complete Figure 4.2: Diagrams that show the relation between the logarithm of the width of a cycle basis, W (B), and the lower bound after 10 seconds of computation of the according MIP for the selected instances and cycle bases, respectively.

134 128 Experiments of widths of cycle bases, does enhance the MIP performance. Moreover, this strong indication is stressed by the following observation. For the Denver Medium 3 network, in 94.4% of the 500 cases, the WFCB heuristic improved the obtained lower bound comparing the two MIPs, the strictly fundamental one and its refined version. For the other two considered instances, Denver Big and Denver Complete, the percentages even amount to 99.4% and 99.8%, respectively. Hence, applying the WFCB heuristic is recommended. Finally, we discuss the initial question: to what extent can we share the observation by Liebchen [Lie06], that the width of a basis suggests how the MIP solver might perform on the according MIP? Liebchen, who investigated a periodic timetabling application using a PESP formulation, correlated the width of a cycle basis to other measures for the computational performance of its MIP. Namely, he considered the LP relaxation value and the total solution time of the MIP. He then observed that for three independent pairs of cycle bases, always the one with a smaller width, leads to a MIP that performs better w.r.t. the LP relaxation value and the total solution time. However, for the previously mentioned reasons, we decided to consider the lower bound of the MIP after 10 seconds of computation as the performance measure for our experiments. Here, an exact correlation between the width of a cycle basis, and the value of its MIP s lower bound could not be observed. Still, by conducting experiments comparing random SFCBs with WFCBs, which, on average, have a far smaller width, the statement of Liebchen can be shared to a certain extend. The reason, why our experiment indicates that correlations are not as explicit as in Liebchen s experiments may be due to the underlying model. Liebchen used an MIP approach for the PESP, whereas we modeled the NSC problem. See Section for problem definitions. The fundamental difference between the PESP and the NSC problem is, that for the PESP, one is given bounds on the periodic tension variables. These bounds then explicitly enter the definition of the bounds of the integer variables for the cycle equations. The bounds, in turn, are essential for the definition of the width of a cycle basis. We conclude this section with the following remark. Remark 4.2. For our implementation of the NSC MIP (1.25), as it is used in Sections 4.1 and 4.3, we used the basis of type B Case Studies In this section we evaluate the NCP model (1.25) on real-world networks of Portland, Section 4.3.3, and Denver, Section Before we do so, however, we will discuss in Section the procedure of evaluating such a model and, in Section 4.3.2, the topic of data acquisition Evaluating an NSC model Although models for practical problems in general or for the NSC, respectively, can be of theoretical value, in most cases the real quality can not be determined until the

135 4.3 Case Studies 129 model is evaluated. However, it is obvious that a new model cannot immediately be used to calculate a coordination for a real traffic network. Rather, significant tests are required. Currently, an appropriate method to test a model for the NSC problem is to use microsimulation, i.e., a simulation software that maps real world traffic behavior as realistic as possible. Among, for example, AIMSUN [FB93], NETSIM [net80, RS90] and TRANSYT, see Section for a brief description, the tool VISSIM 3 is one of the most reputable simulators. At this time it has to be discussed whether a macroscopic NSC model like ours is equitably judged by a microscopic simulation. As already mentioned in 1.3, we consider a situation of high or near saturated traffic volume, in which single vehicles are likely to behave due to coordination like platoons of vehicles. Thus, within simulation such platoon-building behavior should be recognizable. Another issue concerns receiving real-world data: in our case, data of real-world networks on which we can run the simulation. Although the PTV provided us with data of some north american cities, the formating and preprocessing of this data was by no means easy to accomplish. This is discussed in more detail in Section In the following paragraph we report on the simulation environment in VISSIM. VISSIM is described as Microscopic traffic flow simulation for traffic and transit movements [...] [ptv]. In VISSIM we are given a detailed model of the network, which includes the modeling of different driving lanes and an exact modeling of the signalization at each junction. The traffic within the network is organized as follows. Figure 4.3: Screenshot from the microsimulation VISSIM At particular points in the network, the so-called sources a prescribed number of single vehicles, given in vehicles per hour enters the network Poisson-distributed. Then, they follow a prescribed and fixed route until they leave the network at specific points, the so-called sinks. Here, each vehicle is modeled individually. Some quantities, for example small variations in speed, depend on stochastic functions inherent 3 VISSIM abbreviates Verkehr in Städten Simulation.

136 130 Experiments in VISSIM. These stochastic functions are fed with random seeds. It is recommended running the simulation multiple times with different seeds in order to get significant results. For each vehicle, the total duration of its journey from its source to its sink is measured. Here, delay refers to the actual determined journey time minus a best possible journey time. The best possible journey time for a vehicle is the journey time that the vehicle needs for the same route but without any other vehicles on the track and without signalization. That is to say, the measurement delay collects delay due to stops because of signalization and delay due to interaction with other vehicles. However, we think that this VISSIM definition of delay fits the one of the MIP, see Section 1.3. Although microsimulating a particular solution of the NSC in order to evaluate its real-world quality is important, its real value can not be determined until other solutions have been simulated as well. One such solution, which is important to use for a comparison, is the present solution, i.e., the present adjusted offsets in the network. Moreover, simulating, for example, solutions obtained by TRANSYT, see Section 1.1.3, and comparing the simulation results to the ones obtained by our NSC model (1.25) is of great significance. The detailed VISSIM settings that we used for the simulations are collected in the following remark. Remark 4.3. We run the simulation for 3600 seconds (Portland) and 1800 seconds (Denver) with a prior 300 seconds (both) phase to reach a stable situation in the network. We speak of a stable situation when the number of vehicles in the network is nearly constant. For these 300 seconds we do not measure delay. For each solution to simulate, we run the simulation twenty times, each with different random seed, and take the average. We keep the internal VISSIM parameter of calculation frequency, measured in time steps per simulation second, at its default value of Data Acquisition Getting real-world data for testing mathematical models for practical problems is always a problem. This is especially true in the case of instances for the NSC; the required data is substantial: one needs information on the network, on the traffic flow in the network, and last, but not least, on the signalization. Fortunately, the PTV company provided detailed data, i.e., data for the inner-city street networks of Portland and Denver, including information on the traffic flow and signalization. However, this reveals only half of the story. First, the term data has to be specified, since one needs data for the definition of an instance as input for the model and data as it can be used as input for the simulation. Unfortunately, the two situations require different types of data. With input for the simulation one has to know the position of each signal in the network, i.e., its coordinates. On the other hand, this is a useless information for the NSC model, because there we are only interested in, say, adjacency information of the network.

137 4.3 Case Studies 131 So, the question is how to get all the necessary information; or, how can we complete the data? In general, data for real-world traffic networks is acquired by traffic companies, usually by order of public authorities that want some kind of model for their network in order to plan or evaluate a certain traffic scenario. Fortunately, the company PTV provided us with data for some north-american cities. As the PTV is a company that develops traffic-planning software, real-world data for traffic networks is usually preprocessed for and held within their software tools. Namely, the software tools are VISUM 4 and VISSIM. For a brief description of VISSIM see Section VISUM is a Transportation information and planning system for private and public transport, graphical network editing, analysis, evaluation, assignment, forecast and impact calculations [ptv]. In Figure 4.4 the network of Denver, which is discussed in detail in Section 4.3.4, is depicted. (a) VISSIM (b) VISUM Figure 4.4: The street network of Denver as it is modeled in VISSIM and VISUM. Actually, the two views are not independent. Rather, the VISSIM model evolved from the VISUM one by an exportation. As we want to use the microsimulation VISSIM for our evaluation we have to have a VISSIM-model for the network. However, if a VISSIM-model of a network is the only data source available, we face some other problems: within VISSIM the traffic flow is given via inflow definitions and route decision points. As such, an extraction of the relevant information, which is the traffic flow on a link, is complicated. In addition, travel times on the links have to be determined via simulation. Thus, extracting all data just from a VISSIM-model is afflicted with some effort. However, if a VISUM-model of a network is the only data source available, the situation is not optimal either. On the one hand, internal data structures enable a much easier extraction of link flows and travel times. Moreover, a model-export to 4 VISUM abbreviates Verkehr in Städten Umlegung.

138 132 Experiments VISSIM is supported. On the other hand, the exported VISSIM-model often has to be mended by hand. So, at the best, we have a VISUM-model together with a compatible VISSIMmodel, i.e., a VISSIM-model that has copiously been repaired and tested. Fortunately, for the network of Denver, see Section 4.3.4, the PTV provided us with a complete VISUM-model and an associated reworked VISSIM-model. For the network of Portland, see Section 4.3.3, we extracted all necessary information out of an VISSIM-model for the network. Finally, it should be noted that no data for platoon-lengths or saturation rates was given. Instead, we used approved estimations for the platoon lengths, which are specified in the particular sections. For the saturation rate we used a value of 1 vehicle per second Portland The city of Portland is, though not its capitol, with a population of approx. 560,000 (2006) by far the largest city in the U.S. state of Oregon. (a) Aerial view. (b) The network layout. (c) The signals. (d) The signals. Figure 4.5: The inner-city street network of Portland, Oregon. Screenshot from VISSIM. In our study, we consider the downtown part of the inner-city street network of Portland, cf. Fig. 4.5(a). In Fig. 4.5(b) a VISSIM screenshot of this part of the city is displayed. Here, the blueishly highlighted streets indicate bus corridors. Along

139 4.3 Case Studies 133 these corridors traffic responsive signals are installed, whereas everywhere else we find fixed-time control signals. This is illustrated in Fig. 4.5(c). In total, there are 29 traffic responsive signals and 81 fixed-time control ones. For the remaining intersections in the network, either no data about signalization is available or they are simply unsignalized, e.g., the ones along Park Avenue (see the green stripe stretching south west to north east in Fig. 4.5(a)). Although it may not seem so, the VISSIM model of the street network, which we use as database, has the disadvantage in that it has disruptions in the street layout, whereas in the real network are none. See the black lines indicating this disruptions in Figure 4.5(c). Along the aforementioned bus corridor there are traffic responsive signals installed that favor the public transport. The network is subdivided by this corridor and the unsignalized stripe along Park Avenue into separated blocks with fixed-time signals, cf. Figure 4.5(c). We chose the section depicted in Fig. 4.5(d) as the part on which to run our optimization. The square between Taylor St. and Madison St. and between Broadway and 4th Aven. consists of 16 signalized junctions, i.e., 16 fixed-time controlled signals at the intersection. See Figure 4.5(d). Moreover, all signals work with a 54 second cycle length and have a prescribed and fixed signal timing plan. The green phases of the signals have lengths between 17 and 31 seconds. For the transit times on the links a value of 10 seconds is set uniformly, since all links have the same length. Figure 4.6 provides an enlarged view of the chosen network. In this figure we highlight the paths on which the traffic crosses the network. The numbers indicate the corresponding traffic volumes in number of vehicles per hour. Note that the information on paths and volumes has been experimentally extracted from the complete Portland VISSIM model by a large number of simulation runs Figure 4.6: A visualization of the traffic leading paths crossing the network is given. The numbers indicate the amount of traffic measured in vehicles per hour. Due to the given data and conditions, for the optimization, we use the NSC model (1.25) without the constraints for the variable phase sequencing, (1.25c) and (1.25d). Thus, a solution of (1.25) for the Portland instance is a set of op-

140 134 Experiments timal offsets which, because of the network s small size, could be found in less than a second. Therefore, choosing a small cycle basis for the MIP (1.25) was of minor relevance for the Portland network. Table 4.4 summarizes some statistical information on the considered network and its NSC mixed-integer linear program. Table 4.4: Some collected characteristics of the Portland network and the according NSC mixed-integer program Param Portland Network: Number of Nodes 16 Number of Arcs 30 Number of Cycles 15 MIP: Number of Variables 105 Number of Constraints 133 Number of Non-zeros 271 We compare the simulation results of sets of offsets obtained by the following means. As reference value we use the present coordination of the Portland network. Unfortunately, it is not clear how the present coordination was designed. However, we suppose that it was adjusted by a traffic engineer by hand. Further, we decided to include random offsets in our experiments, although we are aware that this comparison is of secondary value. In particular, we consider the average delay in VISSIM of 75 randomly chosen offset sets. Of more interest is the simulation result of the best of these 75 random offset sets since it gives an impression of what can be obtained at the best by blind guessing. Finally, we used offsets obtained by the commercial software tool TRANSYT. We used TRANSYT in version TRANSYT-7F Release See Section for more information. Note that it was necessary to rebuild the network, cf. Fig. 4.5(d), within TRANSYT. The comparison with TRANSYT is crucial in evaluating the performance of a signal network coordination model. The CPU times given in Table 4.5 are to be read Table 4.5: Comparison of delays from different coordinations for the network of Portland Coordination Delay CPU time Relative in s/veh difference Average random % Best random % Present % Optimization 16.1 < 1s 3% TRANSYT s 34% TRANSYT s 4%

141 4.3 Case Studies 135 as follows. For the optimization we used CPLEX 10.1 on a Intel P4 machine with 3.2GHz, 1.0GB RAM running Linux. On the other hand, TRANSYT works under Windows XP on an Intel T43p 2.13GHz with 1.0GB RAM. In the remainder of this section we discuss the results of the simulation given in Table 4.5. A first observation is that taking just any offset, as the average result of the random offset sets shows, is not a good idea. One risks an average delay that is more than twice as big compared to the present coordination. Taking the best of the 75 offsets sets does not yield acceptable results, too. These results already show that there is optimization potential inherent in coordinating a network. One of the main observations from Table 4.5, is that the offsets obtained by our optimization turn out to be better than the present ones. Namely, an optimal solution of our NSC model (1.25) results in a 3% decreased delay value. Although this does not seem like much, one has to note the very short time that was needed to obtain an optimal solution of (1.25): less than a second. On the other hand, obtaining the offsets for the present coordination probably takes much longer. Another important observation from the simulation results is that the quality of offsets obtained by TRANSYT is twofold. Either the offsets are very good, and it took a long time to find them, or they were found in reasonable time, but they are of poor quality. Having both, good offsets found in short time seems impracticable. A short explanation of the two different TRANSYT solutions is required: during the genetic algorithm, TRANSYT evaluates the respective solutions via an internal simulation. For that simulation two modes exist. In the first mode, the singlecycle mode, one single cycle, i.e., once a time period equal to the networks uniform cycle length, is simulated. In the second mode, the so-called multi-cycle, a multiple of cycle lengths is simulated. Obviously, the longer the internal simulation for the evaluation, the more realistic. Hence the results from Table 4.5 are not surprising. The multi-cycle approved offsets are of much better quality than the single-cycle approved ones, 15.9 s/veh compared to 22.2 s/veh. However, this quality gain is achieved at the cost of running time, 800s compared to 10s. All other genetic algorithm parameters have been set equal for both modes, see Remark 1.2. Moreover, the following has to be discussed. Although with our optimization we found an optimal solution, another set of offsets, namely the one by TRANSYT, performed better in simulation. The reason for that is TRANSYT uses a realistic internal evaluation of a solution. That means TRANSYT uses a mesoscopic traffic model, whereas in our MIP we consider the traffic macroscopically. However, when comparing different network coordinations to the present one, we have to admit the following. The present coordination was adjusted for the whole Portland network and not for the 16-signal subnetwork that we selected for our experiments. Hence, such a comparison may appear somewhat unfair. On the other hand, an at-once coordination of the complete Portland network is tricky anyway. This is because different parts of the network, i.e., the blocks separated by the unsignalized corridor or the corridor equipped with traffic responsive signals, do not interact, or, if they do, interact in an unpredictable way. In conclusion observe from Figure 4.6 that the traffic volumes are not too high. Namely, it is not clear whether these values completely justify the assumption of

142 136 Experiments a platoon-like behavior what we named a prerequisite for the applicability of the model. However, since in this network the links between the junctions are rather short, platoons do adjust, which can be observed in the simulation. Also with respect to the NSC model (1.25) itself, we observed during the experiments on the Portland network that the quality of the solution is sensitive regarding the adjustment of platoon-lengths and transit times. That is to say that small changes of the grading of the platoon lengths, as suggested in Section 1.3.5, do affect the solution, i.e., bring different sets of offsets as results Denver As a second real world example network we investigated the inner city street network of Denver. The city of Denver has a population of approx. 560,000 (2005) and is the capitol of the U.S. state of Colorado. In Fig. 4.7(a) an aerial view of downtown Denver can be seen, whereas in Fig. 4.7(b) a VISSIM model of the network is depicted. In total, the network consists of 146 fixed-time controlled junctions with a uniform cycle length of 75 seconds. Unlike in the Portland network, this time we consider the whole network, i.e., we do not tailor a smaller subnetwork. (a) Aerial View. (b) Model of the network. Figure 4.7: The city of Denver, left from aerial view and right in simulation. However, we have to discuss some preparation steps that we had to carry out. The data for the Denver network originated from a VISUM input model. In order to run the simulation, the network with the traffic load had to be exported to VISSIM. Unfortunately, this sometimes caused some incorrect results. For example, connections of streets were not always copied correctly which resulted in slightly different junction topologies. In addition, the original Denver network contained four traffic responsive signals. At these signals, we approximated the traffic responsive controlling by a fixed-time controlling [Nök06], thereby staying very close to the original signal settings. Nonetheless, an advantage of having at hand a VISUM model is that traffic volumes, as they are needed as input for the optimization, can easily be extracted.