Lecture Notes in Economics and Mathematical Systems 603

Transcription

1 Lecture Notes in Economics and Mathematical Systems 603 Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, Bielefeld, Germany Editorial Board: A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten

2 Dirk Briskorn Sports Leagues Scheduling Models, Combinatorial Properties, and Optimization Algorithms 123

3 Dirk Briskorn Department of Production and Logistics University of Kiel Olshausenstrasse Kiel Germany ISBN e-isbn DOI / Lecture Notes in Economics and Mathematical Systems ISSN Library of Congress Control Number: Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper springer.com

4 Basic research is like shooting an arrow into the air and, where it lands, painting a target. Homer Adkins

5 Preface This book is the result of my research on sports leagues scheduling at the Christian-Albrechts-University of Kiel. This research has been done during my employment as research associate at the Chair for Production and Logistics. The challenging research topic as well as the friendly environment made these years enjoyable and pleasant. First of all, I wish to express my gratitude to my thesis advisor, Professor Dr. Andreas Drexl. He has established a very inspiring and motivating research environment, and he always took the time for discussions and helpful advices. Moreover, he refereed this work. Furthermore, I am very grateful to Professor Dr. Sönke Albers for co-refereeing this thesis. Additionally, I would like to thank my colleagues in Kiel for many helpful discussions, comments, and suggestions. I am especially grateful to Dr. Andrei Horbach, Marcel Büther, and Dr. Yury Nikulin. Stefan Wende was a big help regarding the technological background of my work. Ethel Fritz, Jens Heckmann, and Jürgen Lux were also always helpful with most various things. Besides, I thank Professor Frits Spieksma, Juniorprofessorin Dr. Sigrid Knust, and Dr. Thomas Bartsch for stimulating discussions and advice. Finally, I would like to thank my family and, last but not least, Eva for supporting me and bearing me. Kiel, December 2007 Dirk Briskorn

6 Contents 1 Introduction Motivation RelatedWork BasicNotation Outline Basic Problems SingleRoundRobinTournament DoubleRoundRobinTournament r RoundRobinTournament DecompositionSchemes First Schedule Then Break First Break Then Schedule Real World Problems ExternallyGivenConstraints ForbiddenMatches Regions Capacity HighlyAttendedMatches FairnessConstraints Breaks Opponents Strengths Teams Preferences ComputationalStudy GeneratingProblemInstances ComputationalResults Summary... 57

7 X Contents 4 Combinatorial Properties of Strength Groups Factorizations Ordered 1-Factorization of K k,k Ordered 1-Factorizations of K k Ordered Symmetric 2-Factorization of 2K 2k Group-Balanced Single Round Robin Tournaments Group-ChangingSingleRoundRobin Tournaments Complexity Summary Home-Away-Pattern Based Branching Schemes Motivation GeneralHome-Away-PatternSets Achieving Feasible Home-Away-Pattern Sets ChoiceofBranchingCandidates NodeOrderStrategy MinimumNumberofBreaks Achieving Feasible Home-Away-Pattern Sets ChoiceofBranchingCandidates NodeOrderStrategy ComputationalResults GeneralHome-Away-PatternSets MinimumNumberofBreaks Summary Branch and Price Algorithm MotivationandBasicIdea Reformulation SetPartitioningMasterProblem Matching Subproblem BranchingScheme BranchingStrategy NodeOrderStrategy ColumnGeneration Pricing ColumnManagement Lower Bounds Upper Bounds ComputationalResults Summary Conclusions and Outlook...145

8 Contents XI References Index List of Definitions List of Models List of Figures List of Tables...163

9 1 Introduction 1.1 Motivation Sports league scheduling is an essential activity arising in the context of sports events organization. Sports events are of great importance as far as economic aspects are considered. Often, a sports club is a major employer and taxpayer. Thus, private persons as well as public agencies depend on particular sports events as well as on regular sports league seasons. In this research the focus is on regular sports seasons. A sports league schedule (SLS) determines the date and the venue of a match between two opponents. Scheduling must be done by a central agent since each club has specific interests affecting each other. Therefore, a sports league can be seen as a supplier of a season. The customers are composed of fans watching matches in the stadium and tv channels broadcasting them either live or retarded as a summary. Furthermore, sports clubs make money by offering food supply, selling merchandize assortments such as caps, shirts, scarfs, and so on. While the latter highly depends on the current success of a specific club, the former can be supported by constructing pleasant SLSs. Of course, along with attractiveness there are a lot of attributes with respect to the schedules structure, security aspects, resources, and infrastructure to be considered. There are few standard construction schemes for SLSs which all fail if real world constraints are taken into account. It is no surprise that schedules even for professional sports leagues are constructed manually due to the lack of adequate planning tools as reported in Bartsch [5]. Recent research activities have led to several promising approaches, see for example Bartsch et al. [6] and Nemhauser and Trick [68]. However,

10 2 1 Introduction due to the tremendous number of possible SLSs and the sheer difficulty to find even one nearly all existing approaches inspect a rather small part of the solution space. Hence, many potential SLSs are cut out. Another popular approach is to search only one feasible solution neglecting better ones. Thus, there is a great need for efficient scheduling approaches. In this field tackling the whole solution space might be an especially challenging task. Additionally, heuristics findinggood solutions in an acceptable amount of CPU time are of practical interest. 1.2 Related Work There is a vast field of literature concerning sports league scheduling and related topics. In the following we present an overview. There are several common modelling ideas. For example a popular analogy between SLSs and an edge coloring of complete graphs does exist. Consequently, many articles dealing with graph-based models can be found, for example see de Werra [19, 20, 21, 22, 23], de Werra et al. [24], and Drexl and Knust [30]. Brucker and Knust [13] and Drexl and Knust [30] deal with sports league scheduling problems formulated as multi-mode resource constrained project scheduling problems. Note that an edge coloring of a complete graph K n with n 1 colors is equivalent to a 1-factorization of the same graph being closely related to a latin square of size n. A couple of papers study these analogies from a design theory point of view, e.g., Rosa and Wallis [71], Gelling and Odeh [43], Easton et al. [34], and Mendelsohn and Rosa [61]. Several articles concern particular integer programming (IP) formulations, e.g., Bartsch et al. [6], Della Croce and Oliveri [25], and Schreuder [76, 77]. In real world sports leagues schedules with different structures appear. Probably the most popular form is a round robin tournament (RRT). In particular, RRTs attract attention as single RRT (see Bartsch et al. [6], Trick [82], and Easton et al. [33]) and as double RRT. Furthermore, research has been done concerning divisions, e.g., in de Werra [22], and multiple venues not related to the teams, e.g., in de Werra et al. [24]. Most researchers focus on constructing a feasible SLS. Nevertheless, there are approaches evaluating different SLSs and trying to find one having the best evaluation. Among the most popular goals are minimizing the number of breaks, e.g., in Elf et al. [36], Fronček and Meszka [41], and Miyashiro and Matsui [63], minimizing travel costs, see Anag-

11 1.3 Basic Notation 3 nostopoulos et al. [2] and Easton et al. [32], and minimizing carry-overs, see Russell [73]. Many kinds of different sports are considered such as baseball in Russell and Leung [74], basketball in Nemhauser and Trick [68], ice hockey in Fleurent and Ferland [38], soccer in Bartsch et al. [6], de Werra [23] and Schreuder [76], and tennis in Della Croce et al. [26]. Extensive overviews of literature on sports leagues scheduling in the context of operations research are provided by Knust [51] and Rasmussen and Trick [70]. Furthermore, literature can be found on competing strategies for teams, see Machol et al. [58], Gerchak [44] and Ladany and Machol [53] and the motivation for tackling strategy decisions using operations research methods in Mottley [66] and Schutz [79]. Finally, we refer the reader to some articles concerning the economic importance of professional sports for cities or regions in order to emphasize the practical relevance of generating attractive SLSs. Cairns et al. [16], Jeanrenaud [48] as well as Leeds and von Allmen [55] provide extensive surveys taking into account, for example, demand via paid attendances and broadcasts. Furthermore, the relation between sports and economic development is outlined in Baade [3] and Burgan and Mules [14], for example. 1.3 Basic Notation A sports league is a composition of a set T of n, n even, teams competing each other. Competitions can be specified as exactly one team i playing against exactly one other team j and are called matches in the remainder of this work. A match is carried out in exactly one period p out of the set P. A match takes place at one of the both opponents stadiums. Therefore, we can identify a match by a triple (i,j,p). Here, p is the period where teams i and j compete at i s home. As far as not stated otherwise all indices are 1-based. A SLS in general is a timetable determining the time and the venue where a specific match is carried out. Throughout this work, we consider leagues with an even number of teams obeying a RRT structure. The number of teams being even is no restriction to generality since we can add a dummy team if n is odd and, therefore, unconditionally obtain n being even. The matches of RRTs are grouped in a fashion that each team plays exactly once per period. The collection of all matches carried out in a period is called matchday (MD). Hence, a MD consists of n 2 matches. There are several

12 4 1 Introduction different RRT structures but they all have in common that each team i plays against each other team j exactly r times with r N >0. Then, each SLS contains r(n 1) MDs and, consequently, P = r(n 1). 1.4 Outline The work is organized as follows. Chapter 2 is focused on basic problems in sports league scheduling. Different structures for RRTs are presented and corresponding optimization problems are introduced. Furthermore, we provide proofs of complexity. In chapter 3 real world requirements are examined. Moreover, we represent them by means of IP model formulations and provide a computational study. We give detailed insights into consideration of strength groups from a combinatorial point of view in chapter 4. In chapter 5 branching schemes based on a well known decomposition scheme are developed. Furthermore, we provide computational results obtained when employing these branching schemes. Chapter 6 provides an highly flexible exact algorithm to be easily adapted for solving all variations of the problem from chapter 3. Finally, chapter 7 gives conclusions and an outlook on future research. All computational results are obtained using a 3.8 GHz Pentium 4 machine with 3 GBs of RAM. We employ Ilog Cplex 9.0 with default settings (if not mentioned otherwise) as standard solver. Run times are given in seconds.

13 2 Basic Problems The chapter at hand concerns basic problems solely derived from SLSs structural requirements. To this end, we neglect most real world requests in order to focus on complexity induced by the RRT structure itself. Furthermore, we inspect subproblems resulting from popular decomposition schemes. Correspondingly, we define cost minimization problems and give proofs or conjectures for their complexity, respectively. First, we present the well known planar three index assignment problem (PTIAP) which will serve to proof the scheduling problems complexity. Definition 2.1. Given are three sets A,B,C with A = B = C = m, m N, m even, as well as costs d a,b,c for each triple (a,b,c) A B C. Feasible solutions to the PTIAP consist of m 2 triples such that each pair in (A B) (A C) (B C) is contained exactly once. The PTIAP is to find a solution having the minimum sum of chosen triples costs. We give here a formulation of PTIAP as an integer program according to, e.g., Spieksma [80] using m 3 binary variables and 3m 2 constraints. In this formulation y a,b,c equals 1 if triple (a,b,c) is chosen, 0 otherwise. The objective function (2.1) represents the goal of cost minimization. Equations (2.2), (2.3), and (2.4) force each pair to be contained in exactly one chosen triple. In the following we give the decision version of PTIAP which will be referred to as PTIAP-DEC. Definition 2.2. PTIAP-DEC is defined by input and question: Input: Three m-sets A,B,C,andasetD A B C. Question: Does there exist a subset of D containing m 2 triples such

14 6 2 BasicProblems Model 2.1: PTIAP-IP min d a,b,c y a,b,c (2.1) a A b B c C s.t. y a,b,c = 1 a A, b B (2.2) c C y a,b,c = 1 a A, c C (2.3) b B y a,b,c = 1 b B,c C (2.4) a A y a,b,c {0, 1} a A, b B,c C (2.5) that each pair (a,b) (A B), (a,c) (A C), and(b,c) (B C) is contained exactly once in those triples? In Frieze [40] PTIAP-DEC is proven to be NP-complete implying that PTIAP is NP-hard. Complexity of several sports league scheduling problems will be shown by reduction from PTIAP-DEC hereafter. 2.1 Single Round Robin Tournament Single RRTs obey the general structural requests outlined in section 1.3. Additionally, r is specified to be equal to 1 which means that each team meets each other team exactly once. The tournament contains n 1 MDs; see table 2.1 for an instance with n =6.Herei-j denotes team i playing at home against team j. Table 2.1. Single RRT for n =6 period match match match Next, we define the single RRT problem.

15 2.1 Single Round Robin Tournament 7 Definition 2.3. Given a set T, T = n,andasetp, P = n 1,each triple (i,j,p) T T P,i j, represents a match of team i against team j at i s home in period p. Costsc i,j,p are given for each match. A feasible solution to the single RRT problem corresponds to a set of n(n 1) 2 triples such that (i) for each pair (i,j) T T,i < j, exactly one triple of form (i,j,p) or (j,i,p) with p P is chosen and such that (ii) for each pair (i,p) T P exactly one triple of form (i,j,p) or (j,i,p) with j T \{i} is chosen. The problem is to find a feasible solution having the minimum sum of chosen triples cost. Condition (i) implies that each pair of teams meets exactly once while condition (ii) ensures that each team plays exactly once per period resulting in n 1 periods. These requests as well as the goal of cost minimization can be represented as an IP model employing n(n 1) 2 binary variables and 3n(n 1) 2 constraints. Model 2.2: SRRTP-IP min i T c i,j,p x i,j,p (2.6) j T \{i} p P s.t. (x i,j,p + x j,i,p ) = 1 i,j T,i < j (2.7) p P j T \{i} (x i,j,p + x j,i,p ) = 1 i T,p P (2.8) x i,j,p {0, 1} i,j T,i j, p P (2.9) Binary variable x i,j,p isequalto1ifmatch(i,j,p) is carried out, and 0 otherwise. Constraints (2.7) and (2.8) correspond to (i) and (ii), respectively, while (2.6) represents the goal of cost minimization. Cost c i,j,p of a specific match can be seen in a rather abstract way here. For example SRRTP-IP can serve as subproblem of a sports scheduling problem taking into account the real world constraints neglected in SRRTP-IP. Then, c i,j,p might cover, among other components, dual variables. However, there are several further applications of SRRTP-IP having practical relevance:

16 8 2 BasicProblems Teams usually have preferences for playing at home in certain periods, a fact which can easily be expressed through c i,j,p.letpr i,p R be team i s preference to play at home (pr i,p > 0) or to play away (pr i,p < 0), respectively, in period p. A preference pr i,p is stronger than a preference pr i,p if pr i,p > pr i,p. Then, costs can be defined as c i,j,p = pr i,p + pr j,p, for example. Here, cost c i,j,p represents neglected preferences of i and j in p decreased by fulfilled ones if (i,j,p) is carried out. Hence, the objective of SRRTP-IP is to maximize the difference between fulfilled preferences and neglected preferences. Since a major objective of the organizers of a tournament is to maximize attendance we can represent the economic value of the estimated attendance by c i,j,p. Let estimated attendances ea i,j,p be given for each match (i,j,p). Then, we can define costs of SRRTP- IP as c i,j,p = ea i,j,p and obtain the objective to maximize total tournament s attendance. Equivalently, c i,j,p can be defined as the number of seats remaining empty in the stadium of i if (i,j,p) is carried out. Often, a stadium is owned by some public agency and teams have to pay a fee for each match taking place in that particular stadium. This fee might depend on season, day of the week, and time when the match takes place as well as competing events. We can represent it by c i,j,p and obtain the objective to minimize the sum of fees to be paid. A special case of SRRTP-IP arises when the costs are restricted to {0,1}. Thenc i,j,p = 1 denotes that team i cannot play team j in team i s home venue in period p,whereasc i,j,p = 0 denotes that this is possible. A reason for a match (i,j,p) being impossible might be restricted availability of team i s stadium. What we are interested in is to determine whether a feasible schedule, that is, a zero-cost schedule, exists or not. The complexity of the single RRT problem has been independently stated in Briskorn et al. [12] and Easton [31]. The proof in Briskorn et al. [12] is reproduced below for the sake of completeness. In analogy to PTIAP-DEC we define SRRTP-DEC as the decision version of the single RRT problem first. Definition 2.4. SRRTP-DEC is defined by input and question: Input: An instance of single RRT problem having c i,j,p {0,1} for each i,j T, i j, p P. Question: Does there exist a solution having cost equal to 0?

17 2.1 Single Round Robin Tournament 9 Theorem 2.1. The single RRT problem is NP-hard. Proof. We prove theorem 2.1 by presenting a reduction from PTIAP- DEC to the single RRT problem. PTIAP-DEC is proven to be NPcomplete in Frieze [40]. First, we reduce PTIAP-DEC to SRRTP-DEC. We assume, without loss of generality, that m is even. Given an instance of PTIAP-DEC, we now build the instance of SRRT-DEC as follows. There are 2m teams, so we have T = n =2m (and of course P =2m 1). Further, we set { 0 (i,j,p) D, c i,m+j,p = 1 (i,j,p) / D, and 1ifi,j,p {1,...,m},i j, 1ifi,j {m +1,...,2m},i j,p {1,...,m}, 1ifi {m +1,...,2m},j,p {1,...,m}, 1ifi {1,...,m},j {m +1,...,2m}, c i,j,p = p {m +1,...,2m 1}, 0ifi,j {1,...,m},i j,p {m +1,...,2m 1}, 0ifi,j {m +1,...,2m},i j,p {m +1,...,2m 1}, 1ifi {m +1,...,2m},j {1,...,m}, p {m +1,...,2m 1}. This completes the description of the instance of SRRTP-DEC. A yes-answer to the PTIAP-DEC instance corresponds to a yesanswer to the SRRTP-DEC. First, the triples (a,b,c) which constitute the solution of PTIAP-DEC give rise to the following partial solution of SRRTP-DEC: team i = a plays team j = m+b in period p = c at team i s home venue. Since in this way we use only triples from D, wehave ensured that each match between a team i with i {1,...,m}, and a team j with j {m +1,...,2m} is scheduled with zero cost. Second, to schedule the remaining matches, let us first deal with the matches between teams i and j with i,j {1,...,m}, i j. Observethatwe must assign these matches to periods m+1,...,2m 1inordertohave a zero-cost solution. Assigning these matches to m 1 periods can be seen as edge-coloring a complete graph (recall that an edge-coloring of a graph is a coloring of the edges such that adjacent edges have different colors). Indeed, the graph that results when there is a vertex for each of the first m teams, and an edge for each match to be played is complete.

18 10 2 Basic Problems It is well known (see Mendelsohn and Rosa [61]) that, in case m is even as we assumed (m 1) colors suffice to edge color K m. The resulting coloring gives us a feasible assignment of matches to periods (edges with the same color correspond to matches played in the same period). In this way each period receives m 2 matches, each with zero cost. By using the same procedure for teams i and j with i,j {m +1,...,2m}, i j, we find an assignment of the corresponding matches to periods m +1,...,2m 1. Hence, we have found a feasible solution to single RRT problem having total cost equal to zero and, therefore, the answer to SRRTP-DEC is yes. Next, if a zero-cost solution to the single RRT problem exists (which means a yes-answer to SRRTP-DEC), it is not difficult to show that PTIAP admits a zero-cost solution, as well. Indeed, let us focus on the matches between teams i and j with i {1,...,m} and j {m +1,...,2m}. From the construction it is clear that the existence of a zero-cost solution implies that team j never plays at its home venue against team i since this costs 1. Hence, the assignment of matches of team i against team j to periods p, p =1,...,m (which must exist since we assumed that a zero-cost solution to the single RRT problem exists), gives us the solution to PTIAP and, hence, the answer to PTIAP-DEC is yes. Secondly, we reduce SRRTP-DEC to the single RRT problem. This is straightforward, since we can give an answer to SRRT-DEC by solving an instance of the single RRT problem having costs c i,j,p {0,1} corresponding to the costs of SRRTP-DEC. This instance is a special case of the general single RRT problem. Easton [31] proofs NP-completeness of a quite similar problem. The problem, namely SRRTP, is to complete a partial single RRT. Easton [31] shows NP-completeness even if each but three periods are scheduled and each team has no more than three unscheduled matches. This problem can be easily reduced to SRRTP-DEC. The idea is to represent amatch(i,j,p) which is scheduled in the problem in Easton [31] by forbidding all matches (i,j,p),i i,and(i,j,p),j j,insrrtp-dec. Although PTIAP-DEC is not needed for proofing NP-completeness of the single RRT problem the close relation of both problems structures is interesting. We make further use of it in section 2.4. Although the single RRT problem is NP-hard feasible solutions can be found in polynomial time. Well known constructions schemes from graph theory have been adapted. An ordered 1-factorization of the complete graph K n corresponds to a single RRT. See figure 2.1, for example.

19 2.1 Single Round Robin Tournament 11 i 1 i 2 i 3 i 4 Fig Factorization of K 4 The graph s vertices represent teams while edge e represents a match between the teams e is incident to. Edges being in the same 1-factor represent matches in the same MD. Edges in figure 2.1 sketched as dotted, dashed, and solid lines form a MD each. Construction schemes for 1-factorizations of complete graphs are known and can be employed in order to generate single RRTs, see Bartsch [5]. Furthermore, heuristics have been developed which randomly construct 1-factorization within seconds, see Dinitz and Stinson [28] and Hilton and Johnson [46] for example. However, these techniques only cover a quite small part of the solution space and, hence, do not suffice to find optimal or even good solutions. Another result from design theory gives an idea about the difficulties arising while tackling the solution space of single RRT problems. Two 1-factorizations f 1 and f 2 of K n aresaidtobeisomorphicifthereisa mapping φ : V V such that φ(f 1 )=f 2, see Dinitz et al. [29] and Ihrig [47]. Obviously, φ(f 1 ) has the same structure as f 1.Thenumber of classes of non isomorphic 1-factorizations fn c of K n corresponding to single RRTs having different structures is rarely known: f2 c = fc 4 = f6 c =1,fc 8 = 6. Gelling and Odeh [43] state fc 10 = 396 and no less than 20 years later Dinitz et al. [29] report the number for K 12 to be f12 c = 526,915,620. Today fn c = is known neither in general nor for n>12. The number of distinct 1-factorizations fn d is 252,282,619,805,368,320 for n = 12 and is estimated in Dinitz et al. [29] to rise to for n = 18. Further results can be found in Lindner [56]. So, the number of feasible solutions of the single RRT problem is very large and seems to be an excellent example for combinatorial explosion. For problem sizes larger than n = 12 we do not even know the number of solutions of which we are searching for the best one.

20 12 2 Basic Problems 2.2 Double Round Robin Tournament Double RRTs are special cases of RRTs and, therefore, have structures according to the requirements presented in section 1.3. Here, r is set to 2 which means that each team meets each other team exactly twice. The matches are to be carried out at different venues. Consequently, the tournament consists of 2(n 1) MDs. There are different kinds of double RRTs. We distinguish between mirrored double RRTs, double RRTs based on rounds, and (general) double RRTs below. A mirrored double RRT is a double RRT hosting a match between teams i and j, i,j T, i j, inperiodp, p P, ati s home if and only if a match between teams i and j, i,j T, i j,atj s home takes place in period ((t + n 1) mod (2n 2)), that is, the tournament is divided into two single RRTs being complementary to each other; see table 2.2 for an example with n =6. Table 2.2. Mirrored double RRT for n =6 period match match match Most real world sports leagues are scheduled using the mirrored double RRT scheme. Because of its equivalence to single RRTs we will focus on these in chapter 3 where real world requirements are considered. Similar to the concept in section 2.1 we define the mirrored double RRT problem. Definition 2.5. Given a set T, T = n,twosetsp 1 and P 2, P 1 = P 2 = n 1,andcostsc i,j,p for each (i,j,p) T T (P 1 P 2 ),i j, a feasible solution to the mirrored double RRT problem corresponds to a set of n (n 1) triples such that (i) for each pair (i,j) T T,i < j, exactly one triple of form (i,j,p 1 ) or (j,i,p 1 ) with p 1 P 1 is chosen, (ii) for each pair (i,p 1 ) T P 1 exactly one triple of form (i,j,p 1 ) or (j,i,p 1 ) with j T \{i} is chosen, and (iii) for each chosen triple (i,j,p) T T P 1,i j, the triple (j,i,p) T T P 2 is chosen. The goal of the mirrored double RRT problem is to find a feasible solution having the minimum sum of chosen triples cost. Conditions (i) and (ii) ensure a single RRT in P 1 and condition (iii) let the matches in P 2 be complementary to the ones in P 1. Obviously,

21 2.2 Double Round Robin Tournament 13 the mirrored double RRT problem can be solved by solving a single RRT problem where cost c i,j,p are defined as follows: c i,j,p = c i,j,p 1 (p) + c j,i,p 2 (p) i,j T,i j,p P (2.10) Note that p r (p) denotes the period of P r corresponding to index p of P in a single RRT. Theorem 2.2. The mirrored double RRT problem is NP-hard. Proof. We reduce the single RRT problem to the mirrored double RRT problem. The single RRT problem is known to be NP-hard due to theorem 2.1. Given an instance of single RRT problem we construct an instance of mirrored double RRT problem by choosing n = n and setting the costs c i,j,p as follows. { ci,j,p c i,j T,i j,p P 1 i,j,p = 0 i,j T,i j,p P 2 Let t d be an optimal solution to the mirrored double RRT problem. Then, the single RRT t s arranged in P 1 of t d is an optimal solution to thesinglerrtproblem. Suppose there is a single RRT t s having lower cost than t s. Then, we can construct a mirrored double RRT t d having lower cost than t d : set the matches in P 1 according to t s and let the matches in P 2 be complementary to those in P 1.Since t s has lower cost than t s and matches in P 2 do not affect the tournament s overall cost t d has lower cost than t d. The concept of round-based double RRTs is a generalization of mirrored double RRTs. Here, both rounds might not be complementary. The UEFA champions league is a real world example for round-based double RRTs. Table 2.3 illustrates an example with n =6teams. Table 2.3. Round-based double RRT for n =6 period match match match

22 14 2 Basic Problems We define the corresponding round-based double RRT problem in the following. Definition 2.6. Given a set T, T = n,twosetsp 1 and P 2, P 1 = P 2 = n 1,andcostsc i,j,p for each (i,j,p) T T (P 1 P 2 ), i j, a feasible solution to the round-based double RRT problem corresponds to a set of n (n 1) triples such that (i) for each pair (i,j) T T, i<j,exactly one triple of form (i,j,p 1 ) or (j,i,p 1 ) with p 1 P 1 is chosen and (ii) exactly one triple of form (i,j,p) with p (P 1 P 2 ) is chosen, and such that (iii) for each pair (i,p) T (P 1 P 2 ) exactly one triple of form (i,j,p) or (j,i,p) with j T \{i} is chosen. The goal of round-based double RRT problem is to find a feasible solution having the minimum sum of chosen triples cost. Conditions (i) and (ii) imply that each team plays against each other team exactly once in each round (once at home and once away). Condition (iii) assures that each team plays exactly once in each period. We represent the round-based double RRT problem as an IP model employing 2n(n 1) 2 variables and 7n(n 1) 2 constraints. Model 2.3: RBDRRTP-IP s.t. min i T j T \{i} p (P 1 P 2) c i,j,p x i,j,p (2.11) (x i,j,p1 + x j,i,p1 ) = 1 i,j T,i < j (2.12) p 1 P 1 x i,j,p = 1 i,j T,i j (2.13) p (P 1 P 2) j T \{i} (x i,j,p + x j,i,p ) = 1 i T,p (P 1 P 2 ) (2.14) x i,j,p {0, 1} i,j T,i j, p (P 1 P 2 ) (2.15) Equations (2.12) and (2.14) force the arranged matches to form a single RRT in P 1 corresponding to conditions (i) and (iii). Constraint (2.13) ensures that each pair of teams i and j compete twice at different venues representing condition (ii). Hence, taking into account equation (2.14) there is another single RRT formed in P 2.

23 2.2 Double Round Robin Tournament 15 Theorem 2.3. The round-based double RRT problem is NP-hard. Proof. Reduction of single RRT problem to the round-based double RRT problem with n = n can be done exactly as reduction of single RRT problem to the mirrored double RRT problem. A (general) double RRT has no additional requirements compared to those specified in section 1.3 with r set to 2, see table 2.4 for example. Table 2.4. Double RRT for n =6 period match match match We define the corresponding double RRT problem below. Definition 2.7. Given a set T, T = n,asetp, P =2(n 1), and costs c i,j,p for each (i,j,p) T T P,i j, afeasiblesolution to the double RRT problem corresponds to a set of n (n 1) triples such that (i) for each pair (i,j) T T,i j, exactly one triple of the form (i,j,p) with p P is chosen and such that (ii) for each pair (i,p) T P exactly one triple of form (i,j,p) or (j,i,p) with j T \{i} is chosen. The double RRT problem is to find a feasible solution having the minimum sum of chosen triples cost. We represent the double RRT problem as an IP model using 2n(n 1) 2 variables and 2n(n 1) constraints, see (2.16) to (2.19). (2.17) and (2.18) directly correspond to (i) and (ii), respectively. (2.16) states the goal to minimize arranged matches costs. Theorem 2.4. The double RRT problem is NP-hard. Proof. We reduce the single RRT problem to the double RRT problem. The single RRT problem is known to be NP-hard due to theorem 2.1. Given an instance of the single RRT problem we construct an instance of the double RRT problem with n = n as follows. Let f be an arbitrarily ordered 1-factorization of K n, built using the method presented in Schreuder [76] for example. Remember that there is always an 1-factorization of K n if n is even. We define costs c i,j,p of the double RRT problem as follows:

24 16 2 Basic Problems Model 2.4: DRRTP-IP min i T j T \{i} p P c i,j,px i,j,p (2.16) s.t. x i,j,p = 1 i,j T,i j (2.17) p P (x i,j,p + x j,i,p ) = 1 i T,p P (2.18) j T \{i} x i,j,p {0, 1} i,j T,i j, p P (2.19) c i,j,p = c i,j,p i,j T,i j,p {1,...,n 1} M 0 (i,j,n 1+p)with(i,j), i<j, being in 1-factor p of f 0 (j,i,n 1+p)with(i,j), i<j, being in 1-factor p of f otherwise j T \{i} p P c i,j,p. with M = i T Let t d be an optimal solution to the double RRT problem. Then, the single RRT t s arranged in periods {1,...,n 1} of t d is an optimal solution to the single RRT problem. Suppose there is a single RRT t s having lower cost than t s. Then, we can construct a double RRT t d having lower cost than t d :setthe matches in periods {1,...,n 1} according to t s. Additionally, let the matches in periods {n,...,2n 2} correspond to f. Then, matches in periods {n,...,2n 2} have overall cost of zero which, obviously, is the minimum possible value. Clearly, t d is a (general) double RRT and, furthermore, t d has lower cost than t d has since t s has lower cost than t s. 2.3 r Round Robin Tournament An obvious generalization of the RRTs presented so far is to let r N, in particular r>2. We take into account the same interrelations between rounds as outlined in section 2.2. In the following we describe the resulting RRTs and give corresponding cost minimization problems

25 2.3 r Round Robin Tournament 17 all of which are NP-hard. We renounce to give proofs of complexity and IP models, respectively, since they are straightforward from those in section 2.2. A mirrored r-rrt is a r-rrt hosting a match between teams i T and j T, i j, ati s home in period p {1,...,(r 1)(n 1)} if and only if a match between teams i and j at j s home takes place in period p + n 1. Hence, the tournament is divided into r single RRTs where single RRT r, r {1,...,r 1}, is complementary to single RRT r + 1. We define the corresponding cost minimization problem hereafter. Definition 2.8. Given a set T with T = n,setsp r with P r = (n 1) for each r {1,...,r}, andcostsc i,j,p for each (i,j,p) T T ( r r =1 P ) r,i j, a feasible solution to the mirrored r-rrt problem corresponds to a set of r n (n 1) 2 triples such that (i) for each pair (i,j) T T,i < j, exactly one triple of form (i,j,p 1 ) or (j,i,p 1 ) with p 1 P 1 is chosen, (ii) for each pair (i,p 1 ) T P 1 exactly one triple of form (i,j,p 1 ) or (j,i,p 1 ) with j T \{i} is chosen, and (iii) for each chosen triple (i,j,p) T T P r, i j, r {1,...,r 1}, the triple (j,i,p) T T P r +1 is chosen. The goal of the mirrored r-rrt problem is to find a feasible solution having the minimum sum of chosen triples cost. Conditions (i) and (ii) ensure that a single RRT takes place in the first round. Condition (iii) induces each round to be complementary to the previous and the following one. Obviously, mirrored r-rrt problems can be solved by solving single RRT problems. To this end, we set cost of a single RRT problem according to a generalization of equation (2.10): c i,j,p = r 2 r =1 2 r c i,j,p 2r 1 (p) + r =1 c j,i,p 2r (p) i,j T,i j,p P (2.20) A round-based r-rrt is a r-rrt according to characteristics introduced in section 1.3. Its periods can be partitioned into r rounds where each round is a single RRT. We define the corresponding cost minimization problem below. Definition 2.9. Given a set T with T = n,setsp r with P r = (n 1) for each r {1,...,r}, andcostsc i,j,p for each (i,j,p) T T ( r r =1 P ) r,i j, a feasible solution to the round-based r-rrt

26 18 2 Basic Problems problem corresponds to a set of r n (n 1) 2 triples such that (i) for each r {1,...,r} the chosen triples (i,j,p) with p P r form a single RRT and such that (ii) for each pair (i,j) T T,i j, atleast r 2 triples (i,j,p) with p r r =1 P r are chosen. The goal of the round-based r-rrt problem is to find a feasible solution having the minimum sum of chosen triples cost. Condition (i) is stated straightforwardly. Condition (ii) limits the difference of number of matches between teams i T and j T, i j, at i s home and number of matches between teams i and j at j s home to be no more than 1. A(general)r-RRT is fully specified by the characteristics given in section 1.3. In the following, the cost minimization problem is given. Definition Given a set T with T = n,asetp with P = r(n 1),andcostsc i,j,p for each (i,j,p) T T P,i j, afeasible solution to the r-rrt problem corresponds to a set of r n (n 1) 2 triples such that (i) for each pair (i,j) T T,i j,atleast r 2 triples (i,j,p) with p P are chosen, such that (ii) for each pair (i,j) T T,i < j, exactly r triples (i,j,p) or (j,i,p) with p P are chosen and such that (iii) for each pair (i,p) T P exactly one triple of form (i,j,p) or (j, i, p) with j T \{i} is chosen. The goal of r-rrt problem is to find a feasible solution having the minimum sum of chosen triples cost. Condition (i) limits the difference of number of matches between teams i T and j T, i j, ati s home and number of matches between teams i and j at j s home to be no more than 1. Condition (ii) implies that each pair of teams meets exactly r times. Condition (iii) assures that each team has exactly one opponent per period. Note that we can drop (ii) if r is even, since then (ii) is implied by (i). 2.4 Decomposition Schemes Due to the complexity of RRT problems outlined in sections 2.1 to 2.3 two decomposition schemes are used frequently. Both separate the decisions about the period a match takes place in and about the venue it is carried out at. First Schedule Then Break: First, each pair of teams is fixed to compete in a specific period. Based on this timetable each match s venue is determined, see Trick [81] for example.

27 2.4 Decomposition Schemes 19 First Break Then Schedule: First, the matches venues are decided. Afterwards, the matches are assigned to periods, see Nemhauser and Trick [68] for an example. Optimization problems considered below refer to single RRTs. However, adaption to other basic problems is straightforward First Schedule Then Break An opponent schedule, as defined in Post and Woeginger [69], is a timetable which determines for each pair (i,p), i T, p P,theopponent of team i in period p. See table 2.5 for an example corresponding to the single RRT in table 2.1. Team i s opponent in period p is specified in the row corresponding to i in column p. Table 2.5. Opponent Schedule for n =6 team In the following, we formally define the problem to find the minimum cost opponent schedule. Definition Given a set T, T = n,asetp, P = n 1, and costs c i,j,p for each (i,j,p) T T P, i<j, a feasible solution to the opponent-schedule problem corresponds to a set of n (n 1) 2 triples such that (i) for each subset of teams (i,j) T T, i<j,exactly one triple (i,j,p) with p P is chosen and such that (ii) for each pair (i,p) T P exactly one triple of form (i,j,p), j T, i<j,or(j,i,p), j T, j<i, is chosen. The goal of the opponent-schedule problem is to find a feasible solution having the minimum sum of chosen triples cost. Condition (i) ensures that each team meets each other team while condition (ii) forces each team to compete exactly once per period.

28 20 2 Basic Problems These requests as well as the cost minimization goal can be represented as an IP model employing n(n 1)2 2 binary variables and 3n(n 1) 2 constraints, see (2.21) to (2.24). Model 2.5: Opponent-IP min c i,j,px i,j,p (2.21) i T i<j p P s.t. x i,j,p = 1 i,j T,i < j (2.22) p P x i,j,p + x j,i,p = 1 i T,p P (2.23) j<i i<j x i,j,p {0, 1} i,j T,i < j,p P (2.24) Binary variable x i,j,p, i,j T, i<j, p P, isequalto1ifand only if teams i and j compete in p. The objective to minimize the cost corresponding to the opponent schedule is given in (2.21). Restriction (2.22) assures that each pair of teams meet and constraint (2.23) forces each team to play exactly once per period. Theorem 2.5. The opponent-schedule problem is NP-hard. Proof. We reduce the single RRT problem to the opponent-schedule problem by setting costs c i,j,p as follows: c i,j,p =min{c i,j,p,c j,i,p } i,j T,i < j,p P Givenanoptimalsolutiont o to the opponent-schedule problem we can construct an optimal solution t s to the corresponding single RRT problem as follows: Foreachchosen(i,j,p), i<j, of the opponent-schedule problem choose (i,j,p) of the single RRT problem if c i,j,p c j,i,p. Foreachchosen(i,j,p), i<j, of the opponent-schedule problem choose (j,i,p) of the single RRT problem if c i,j,p >c j,i,p. Obviously, t s is a single RRT. Furthermore, t s is optimal to the single RRT problem. Suppose there is a solution t s to the single RRT problem having lower cost than t s. Then, there is a solution t o := {(i,j,p) i<j,(i,j,p) t s (j,i,p) t s } to the opponentschedule problem having lower cost than t o.

29 2.4 Decomposition Schemes 21 In the course of the first-schedule-then-break scheme the venue of each match is determined next. The venue can be specified by assigning a home-away-pattern (HAP) to each team. Definition A HAP is a string of length n 1 containing 0 at slot p if the specific team plays at home in period p, 1 otherwise. Definition A HAP set is a collection of HAPs where exactly one HAP is assigned to each team. Table 2.6 illustrates the HAP set corresponding to the single RRT shown in table 2.1. Table 2.6. HAP set for n =6 team Let h be a HAP set and let h i,p be the entry corresponding to team i T and period p P. Then, h is called feasible for an opponent schedule o if for each team i competing team j in period p according to o entries h i,p and h j,p are not identical. Note that the HAP set presented in table 2.6 is feasible to the opponent schedule given in table 2.5. Probably, given an opponent schedule the most popular goal when determining the matches venues is to find the HAP set which is feasible to the opponent schedule and has the minimum number of breaks. A break occurs if a team plays twice at home or twice away in two consecutive periods. The resulting break-minimization problem is focus of many research activities. It is known from de Werra [19] that the number of breaks can not be less than n 2. Elf et al. [36] conjecture the break-minimization problem to be NP-hard in general. Miyashiro and Matsui [63] and Miyashiro and Matsui [62] proof the problem of either finding a HAP set with the minimum number of n 2breaks if one exists or deciding there is no such HAP set to be solvable in polynomial time. The same is true for equitable HAP sets assigning exactly one break to each team.

30 22 2 Basic Problems Post and Woeginger [69] consider partial opponent schedules containing less than n 1 periods. They state that the break minimization problem for partial opponent schedules is NP-hard if the number of periods is greater than First Break Then Schedule First, for each team i T and each period p P it is decided whether i plays at home or away in p. Reasonably, a HAP set can be constructed considering the preferences of teams to play at home or away, for example. Furthermore, stadium availability and fixed matches can be incorporated. The resulting HAP set restricts the following scheduling process: no team playing at home or away in period p according to the HAP set can play away or at home, respectively, in the final SLS. Note that, consequently, two teams i T and j T can not compete in period p if h i,p = h j,p. Obviously, there is no guarantee that a random HAP set h {0,1} n (n 1) allows one single RRT to be arranged. In the remainder, a HAP set h is called feasible if and only if at least one RRT can be scheduled based on h. We define the corresponding decision problem. Definition We define the HAP set feasibility problem as follows: Input: A HAP set h {0,1} n (n 1). Question: Is h feasible? There are two obvious necessary conditions for a HAP set h to be feasible: (i) HAPs of two teams can not be identical. If two HAPs are identical the corresponding teams can not play against each other in any period. (ii)each column of h has to contain exactly n 2 zeros.ifthisdoesnot hold for a specific period p not each team can play in p. Unfortunately, (i) and (ii) are not sufficient for h to be feasible. Miyashiro et al. [64] provide additional necessary condition (2.25) for HAP sets to be feasible. Let c 0 (T,p)andc 1 (T,p)bethenumberof zeros and ones, respectively, corresponding to a subset T T of teams and column p. min { c 0 (T,p),c 1 (T,p) } ( T ) 0 T T (2.25) 2 p P

31 2.4 Decomposition Schemes 23 Condition (2.25) states that each subset T of teams must be allowed to play ( T ) 2 matches against each other. The number of matches between teams of T in each period is restricted to the minimum of numbers of teams in T playing at home and away, respectively. Note that condition (2.25) is a generalization of (ii). As shown in Miyashiro et al. [64] for HAP sets having the minimum number of breaks condition (2.25) can be checked in polynomial time and, moreover, it is conjectured to be sufficient. However, sufficiency is not proven and, therefore, complexity of the HAP set feasibility problem is open so far for arbitrary HAP sets as well as for HAP sets having minimum number of breaks. We propose a condition for HAP sets to be feasible which can be checked in polynomial time. First, we formulate IP model HAP-setfeasibility in order to check feasibility of a given HAP set h as proposed in Briskorn [8]. Again, binary variable x i,j,p, i,j T, j<i, p P, is equal to 1 if and only if teams i and j compete in period p. Model 2.6: HAP-set-feasibility maxz h = i T x i,j,p (2.26) j T,j<i p P s.t. p P x i,j,p 1 i,j T,j < i (2.27) j T,j<i x i,j,p + j T,j>i x j,i,p 1 i T,p P (2.28) x i,j,p h i,p h j,p i,j T,j < i,p P (2.29) x i,j,p {0, 1} i,j T,j < i,p P (2.30) Objective function (2.26) represents the goal to maximize the number of matches while constraints (2.27) and (2.28) assure a single RRT. More precisely, constraint (2.27) forces each pair of teams to meet at most once while constraint (2.28) restricts the number of matches per team and period to be less than or equal to one. The entry of HAP set h corresponding to team i and period p is denoted by h i,p. Then, (2.29)

32 24 2 Basic Problems takes care of the HAP set h such that no pair of teams can meet in a period where both of them play at home or away, respectively. Clearly, h is feasible if and only if z h = n(n 1) 2.Let z h be the maximum objective value to the linear programming (LP) relaxation of HAP-set-feasibility. Then, z h z h and, hence, z h = n(n 1) 2 is a necessary condition for h to be feasible. Furthermore, we can decide in polynomial time whether z h = n(n 1) 2 or not, see Garey and Johnson [42]. Theorem 2.6. Condition z h = n(n 1) 2 is strictly stronger than (2.25). Proof. First, we show that each HAP set h being infeasible according to condition (2.25) is infeasible according to condition z h = n(n 1) 2, as well. Suppose z h = n(n 1) 2. We count the matches between teams in T T in period p as z T h,p. Obviously, p P zt h,p = T ( T 1) 2. Moreover, min{c 0 (T,p),c 1 (T,p)} z T h,p and, thus, p P min {c 0(T,p),c 1 (T,p)} ( T ) 2 holds. Second, we consider a HAP set h provided by Kashiwabara [49] with n = 14 shown in table 2.7. HAP set h fulfills (2.25) but has z h =90< 91 = n(n 1) 2. Table 2.7. Infeasible HAP set for n =14 team