A polynomial approach to the fair multi-flow problem

Transcription

1 A polynomial approach to the fair multi-flow problem Dritan Nace and Linh Nhat Doan Laboratoire Heudiasyc UMR CNRS 6599, Université de Technologie de Compiègne, Compiègne Cedex, France Abstract In this paper we consider the fair multi-flow problem in capacitated networks. We present an iterative algorithm and show the polynomiality of this approach. It is a linear programming based approach which allows a lexicographical maximization of the vector of individual flows in multi-commodity networks. Theoretical analyses of this problem are described, and an application study to telecommunication networks is presented in detail. 1 Introduction Flow problems in networks have been extensively studied since the seminal work of Ford and Fulkerson in this area ([7]). A number of variants of the flow problem in a graph with edge capacities frequently arise in operations research and other fields (see [1]). A widely studied problem is flow maximization. In ([7]), the authors have shown that multiple-source multiple-sink networks can be reduced to single-source single-sink networks by the adjunction of a super-source, a super-sink and some additional arcs. As a consequence, the problem can be solved by the well-known labeling method for searching augmenting paths. However, this is not applicable when, instead of maximizing the flow globally, one needs to distribute it fairly. This latter problem, which is the main scope of the paper, has applications in numerous economic application settings like telecommunications, (e.g. the equitable distribution of resources among a number of sessions or users). Generally speaking, it might be desirable to maximize, with respect to network resources, the minimum amount which is supplied to sinks or commodities (source-sink couples). Furthermore, it is sometimes a requirement to extend the expected fairness beyond this basic level to a more This work is supported by France Telecom R&D. 1

2 globally fair distribution of resources. We consider here the lexicographical maximization of the vector of individual flows in multi-commodity networks, called in the sequel the fair multi-flow problem. Related work: Several variants of the fair (multi)flow problem have been presented and analyzed in [17, 3, 1, 13, 10, 8, 6, 19]. The first one studied in the literature is the fair flow problem in a single-source multi-sink 1 network. This problem can be stated as follows: in a given capacitated network, find the maximal flow from a given source to a set of sinks, such that the amount of flow supplied to sinks be as fair as possible. One can distinguish here two quite different cases: splittable and unsplittable fair flows with respect to the number of paths used to transfer the flow (each flow supplied to a sink is transported through a single (resp. multiple) path for the unsplittable (resp. splittable) case). In [17], the author has put forward an elegant method for the fair single-source splittable flow problem. Conversely, the fair singlesource unsplittable flow problem is shown to be NP-complete: special cases of the latter problem include several fundamental load-balancing problems, (see [13]). However, in many application contexts (e.g. transport), problems are generally not limited to single-source networks: several commodities are required to share the same underlying network. In this paper we study one such model, known as the multi-commodity flow problem, where network resources have to be distributed in the fairest way among commodities, each of which is specified by a source and a sink. Notice that we are not limited to the problem of maximizing the minimum flow among commodities, but we study a more general problem, that is, finding lexicographically maximal multi-flows with respect to given commodities. We call this problem the Fair Multi-Flow problem and we will formally define it in the next section. Issues on fairness are a key concern in numerous studies inspired from applications in the telecommunications area such as congestion and flow control, load-balancing or fair routing 2. A majority of these works have focused on the problem of how to associate the rates with a set of given connections (sessions/users), and very little work has been devoted to understanding the relationship between the way the routing paths (of the connections) are constructed in one hand and the resulting fairness quality and throughput in the other hand. Even when both source-rate and route computation are considered [13, 10, 6, 15], these works are either limited to a given set of paths, or the authors are contented to give a heuristic or provide framework of approximation algorithms. Furthermore, to the best of our knowledge, there is no approach which resolves optimally the fair multi-flow problem in capacitated networks, which is the objective of this work. 1 The problem is quite similar for multi-source single-sink networks. 2 A brief summary of these works is given in section [4]. 2

3 Contribution of this work: In this paper we introduce a polynomialtime algorithm for the fair multi-flow problem in capacitated networks. It is a linear programming based approach that maximizes lexicographically the vector of individual flow values. We also investigate some theoretical properties relative to fair multi-flows, and give an example of an application to elastic traffic routing in telecommunication networks. This work, however, is limited to the splittable flow case, (the unsplittable one is shown to be NP-complete). The paper is organized as follows: in section 2 we provide a mathematical framework and introduce an iterative (polynomial) algorithm for achieving an optimally fair splittable multi-flow in a capacitated network. Next, in section 3, we examine the theoretical properties and also provide proof of the optimality of our approach. Section 4 is devoted to some applications in telecommunication networks. In section 5 we report some numerical results and compare fairness and throughput objectives. Finally, in section 6 we conclude and briefly outline orientations for future work. 2 Mathematical modeling and resolution approach In this section we introduce an approach for the fair multi-flow problem. To start with, let us give some definitions and notation useful for the remainder of the paper. 2.1 Definitions and notation A capacitated network is a given triplet N = (V, A, C), where V is a nonempty finite set whose elements are the nodes. A is the set of ordered pairs (called arcs) of nodes, and C is a function from A to nonnegative reals, called capacity function. For each arc (i, j), the associated capacity is denoted by C ij. Let D be the set of commodities d labeled in {1, 2,..., D }. Each commodity has a source node s d and a sink node t d, and eventually a target value or weight. A flow f d (of value b 0), in network N with respect to commodity d is a function from A to R + such that: (i, j) A fij d C ij (1) i V fij d b, i = s d fji d = b, i = t d (2) j:(i,j) A j:(j,i) A = 0, otherwise where fij d give the arc-flow values. Each flow satisfying constraints (1) and (2) is called feasible. A multi-flow f is composed of feasible flows f d with respect to commodities d D such that: (i, j) A, fij d C ij. This d D 3

4 multi-flow is called feasible. The theorem of decomposition ([1]) suggests that a flow can be decomposed in a set of tuples (path, carried value) and a multi-flow in a set of flows, whose aggregation satisfies capacity constraints. Definition 1 Given a multi-flow f, the vector λ = {λ[d], d D} whose coordinates give the flow value with respect to commodities, is called the flow vector of multi-flow f. - Given a flow vector λ, let λ denote the vector with the same coordinates as λ, but arranged in order of increasing magnitudes. Thus, generally λ[k] λ [k]. - Let F denote the set of all the feasible multi-flows in network N with respect to a set of commodities D. Definition 2 A multi-flow f, whose flow vector is λ, is called optimally fair if for every f F with flow vector λ, λ is lexicographically greater or equal to λ. Similarly, we define the flow vector of an optimally fair multi-flow as optimally fair. An important property of an optimally fair multi-flow is that it is not possible to increase the value of any commodity flow exclusively at the expense of flows whose values are greater, (see [3, 14]). Mathematically, this can be stated as follows: Given a flow vector x, where x[d] gives the value of flow d, we call x feasible if there is a feasible multi-flow satisfying it. Then, the flow vector x is said optimally fair if it is feasible and for any other feasible flow vector y, such that d D : y[d] > x[d], than p D : y[p] < x[p] x[d]. This definition is derived from the max-min fairness definition given in [3] for the fixed single path routing case. At this stage we assume that commodities are typically without priorities, but the above fairness definition can be easily generalized by using weights. In the sequel, by the term fair multi-flow we mean invariably an optimally fair splittable multi-flow. Similarly, we use fair flow vector for optimally fair (splittable) flow vector. Now, the fair multi-flow problem can simply be stated: given a capacitated network N(V, A, C), find a fair multi-flow f with respect to a given set of commodities D. 2.2 Resolution approach We introduce here an algorithm for resolving the fair multi-flow problem defined above. Obviously, the induced flow vector must be lexicographically maximal. This brings us to the main idea behind the algorithm: computing consecutively some multi-flow such that its flow vector is lexicographically maximal for the first ordered k coordinates, and go on like this until the flow vector be totally lexicographically maximal. 4

5 In the sequel an arc will be referred as saturated when there is no more capacity left on it, that is, all its capacity has already been assigned to flows. Some flow/commodity is called saturated when we cannot increase its value while satisfying a given set of requirements, (related to fairness). - Let D k be the set of commodities saturated at the same value λ [k]. The algorithm can then be stated as follows: Algorithm FMF, (Fair Multi-Flow) Given: A capacitated network N(V, A, C) and a set of commodities d D. Required: Find the fair multi-flow and the associated flow vector. (I) Set k:=1; (II) While d D not yet saturated, do: - Solve the problem P k ; (Resolve a LP problem (see below) and compute the current λ [k] value.) - Compute D k ; (Find the set D k of commodities which are definitely saturated at current λ [k].) - p [k + 1, k + D k ] set: λ [p] := λ [k]; - d D k set: λ[d] := λ [k]; - Set k := k + D k ; (III) The flow vector obtained at the last step is lexicographically maximal, and the obtained multi-flow is thus a fair one. where Solve the problem P k and Compute D k are both procedures detailed below. Obviously, each step of the algorithm is intended to maximize the current λ [k]. It terminates when all commodity flows are saturated. Formulating and solving the problem P k : Obviously, the problem P k can be modeled as a multi-flow problem aiming to achieve some level of fairness. We have modeled this problem through a classic arc-node LP formulation as follows: (i, j) A i V, d D p:1 p<k i V, d D \ ( 1 p<k D p ) j:(i,j) A j:(i,j) A f d ij f d ij j:(j,i) A j:(j,i) A Max λ [k] fij d C ij (3) d D λ[d], i = s d fji d = λ[d], i = t d (4) = 0, otherwise λ [k], i = s d fji d λ [k], i = t d (5) = 0, otherwise (i, j) A, d D f d ij 0 (6) 5

6 where constraints (3) are capacity constraints and constraints (4 and 5) are mass balance constraints. Finally, constraints (6) indicate that arc-flow values must be non-negative. Procedure Compute D k : In practice, when several commodity flows are saturated at the end of the k th step (let us call the corresponding set Q), we cannot distinguish those which are really saturated (D k ) from others (see next section, Lemma 1). Once the corresponding λ [k] has been computed, it will be considered as fixed during the procedure. Then, we search for D k as follows: Compute D k (I) Set D k := Q. (Q is obtained when solving problem P k.) (II) For q Q do: Solve the current problem D k q. (The D k q problem is a slightly modified version of P k and it is modeled through a like ɛ-formulation, (see below). At the end of the procedure we get ɛ.) If ɛ > 0, set D k := D k \ q. (III) At the end of the procedure we get D k which is the set of newly saturated commodity flows to given λ [k]. The current problem D k q is given by: i V, d D \ ( (i, j) A i V, d D p:1 p<k 1 p<k D p ), d q i V j:(i,j) A j:(i,j) A j:(i,j) A f d ij f d ij f q ij j:(j,i) A j:(j,i) A j:(j,i) A Max ɛ fij d C ij (7) d D λ[d], i = s d fji d = λ[d], i = t d (8) = 0, otherwise λ [k], i = s d fji d λ [k], i = t d (9) = 0, otherwise λ [k] + ɛ, i = s q λ [k] ɛ, i = t q (10) = 0, otherwise (i, j) A, d D f d ij 0 (11) Obviously, all commodity flows contained in D k are saturated and are thus attributed the flow value λ [k]. We recall that λ [d] does not denote the flow value of commodity d. The latter value depends on the set D k where the commodity d is contained, (i.e. d D k implies that the value associated with commodity flow d takes λ [k]). 6 f q ji

7 In the sequel we call first-level fair multi-flow, each multi-flow satisfying the constraints of problem P 1. An important property of these multi-flows is that they maximize the minimum value of commodity flows. Thus, each commodity flow is assured to have at least as much as the minimum of maximum flows that could be offered to commodities in D. 3 Some theoretical properties In this section we study the optimality and polynomiality of our approach, and we highlight some properties of fair multi-flows and the fair flow vector. We recall that the existence and uniqueness of the fair allocation vector (similar to flow vector) for the fixed single path flow case is already given in [3, 14]. In the following we extend these results for the dynamic multipath flow case and state and prove some characteristics of the fair flow vector. The starting point for this is showing how to build some fair multi-flow. Thus, the primary objective of this section is to prove that the FMF algorithm given above enables the fair multi-flow computing in polynomial time for a given capacitated network and set of commodities. This will also permit to show the uniqueness of the fair flow vector 3. Let us begin by considering the correctness of our iterative algorithm, which is not obvious. Indeed, distinct multi-flows can satisfy all constraints for problem P k at each step k of the algorithm. Furthermore, several commodity flows might possibly be simultaneously saturated. Recall also that at each step of the FMF algorithm we get some multi-flow f and the k th coordinate value of the vector λ (whose coordinates are these of the flow vector of multi-flow f arranged in order of increasing magnitudes). Once computed, this value, (λ [k]), is fixed for the corresponding commodity flow(s) in D k, and used as a constant during the remainder of the calculations. In the following, (Theorem 1), we prove formally the optimality (in the sense fair) of the multi-flow computed by our FMF algorithm. For this, we need to state and prove some preliminary results: Lemma 1 All multi-flows obtained at the end of each step of the FMF Algorithm: a) are saturated simultaneously in a non-empty unique set of commodities, b) are saturated simultaneously in a non-empty unique set of arcs. a) Let us consider the set of multi-flows satisfying all constraints associated with the problem P k, (step k). Obviously, for each multi-flow there is at least one newly saturated commodity flow. Let us denote as D k the intersection of all these sets of newly saturated commodity flows with respect to all possible multi-flows obtained as solution of problem P k. D k is 3 However, there is no uniqueness for fair multi-flows. 7

8 not empty. Indeed, without loss of generality, let us suppose by absurdity that there exist two multi-flows Φ 1 and Φ 2 satisfying all constraints with respect to P k, and having quite distinct set of saturated flows at the same value λ [k]. These multi-flows are such that Q 1 D \ ( 1 p<k D p) (resp. Q 2 D \ ( 1 p<k D p) represents the set of newly saturated commodity flows for Φ 1 (resp. Φ 2 ) and Q 1 Q 2 =. Let us set Φ = (Φ 1+Φ 2 ) 2. It is obvious that multi-flow Φ satisfies capacity and mass-balance constraints associated with the problem P k. Furthermore, it is not saturated in any commodity flow contained in D \ ( 1 p<k D p). In these conditions, we increase at the same pace the flow value to each fij d with respect to non-saturated commodities (d D \ ( 1 p<k D p)) until the network becomes saturated. Obviously, λ [k] will be increased and it will get a strictly superior value than that obtained by Φ 1 and Φ 2, which contradicts the fact that the latter ones are obtained as solution of the problem P k during the k th step. We have consequently proved the first part of the above lemma. b) The second point can be proved in a similar way. It will be noted that each multi-flow solution of problem P k necessarily saturate some arcs on the graph. Following the same idea as above, we can state that the intersection of all these sets of arcs is necessarily not empty, and unique. It follows that in all multi-flows obtained as solution at the end of the k th step, a unique set of commodities (D k ) is saturated to some given set of arcs, called S k. The result of the above lemma can be generalized as follows: Lemma 2 All multi-flows obtained at the end of the FMF Algorithm saturate commodities and arcs in a given unique order. In the view of the FMF algorithm, it can easily be seen that commodities are saturated in a given order, since this is fixed through the consecutive steps of the algorithm. However, this is not obvious when considering the order of saturating arcs. We will prove it in the following by mathematical induction on the number of steps, that is: each multi-flow solution obtained at the end of the k th step saturates arcs consecutively in the order fixed by sets S p, (1 p k). According to Lemma 1, this assumption is verified for k = 1. Let suppose now that it holds at the end of the k th step and prove it for the next coming step. Indeed, according to Lemma 1, all multi-flow solution obtained at the end of the (k +1) th step, saturate arcs at the same unique set S k+1. Obviously, such multi-flows are solution for the preceding problem P k and thus verify the assumption given above, that is, these multi-flows saturate the arcs consecutively in the order of given sets S p:1 p k. Consequently, each multi-flow obtained at the end of the k + 1 th step saturates arcs in a given unique order fixed by consecutive sets S p:1 p k, which ends the proof of the lemma. Remark: The commodities (resp. arcs) contained in the same D k (resp. 8

9 S k ), are interchangeable in the saturating order since they saturate simultaneously at the same value λ [k]. We can now state and prove an immediate result concerning the number of steps of the FMF Algorithm. Lemma 3 The FMF Algorithm terminates in at most Min{ D, A } steps. According to Lemma 1, at each step of the algorithm a non empty set of arcs and commodity flows is saturated. So we cannot have more than A or D distinct D k, and consequently the number of steps cannot exceed A or D. Now, the optimality of the multi-flow obtained at the end of the algorithm can be deduced as follows: Theorem 1 The multi-flow obtained at the end of the algorithm is a fair one. For this, we have to prove that the flow vector obtained at the end of the algorithm is lexicographically maximal. Let us denote with Γ (resp. Λ) the fair multi-flow (resp. multi-flow obtained by FMF), and γ (resp. λ) the corresponding flow vector. As noted in the previous section, γ and λ give the respective flow vector (of γ and λ) reordered in increasing order of magnitudes. Let suppose by absurdity that k such that γ [k] > λ [k] and l < k, we have γ [l] = λ [l]. It can easily be verified that the multi-flow Γ satisfies all constraints of the problem P l for every l [1, k 1]. According to the Lemma 1 and 2, this proves that it is saturated in all commodities contained in consecutive sets D l at λ [l]. Consequently, Γ also satisfies all constraints of the problem P k, and it could not give a better bound for λ [k], which proves that γ [k] = λ [k] and ends the proof of the theorem. The later result shows also that γ = λ and consequently the uniqueness of the vector γ. Combining the result of the above theorem and this of Lemma 2, one can easily verify that γ = λ. Indeed, some fair multi-flow satisfies necessarily all constraints of the problem at the end of the algorithm, and consequently its flow vector is the same to this obtained by the algorithm FMF. This can be formally stated as follows: Theorem 2 Given a capacitated network and a set of commodities, the fair flow vector is unique. In the following we state and prove the polynomiality of our approach. Theorem 3 The fair multi-flow problem can be solved in polynomial time. The FMF Algorithm computes the fair flow vector and its corresponding fair multi-flow in at most Min{ D, A } steps, and each step is composed of at most D LP problems (taking into account the procedure Compute D k ). 9

10 Then, the proof of the theorem follows from the fact that each LP problem is stated as a classic multi-flow problem which is known to be polynomial-time, (Khachian s ellipsoid algorithm, Karmarkar s interior-point algorithm). Some immediate corollaries follow: Corollary 1 The fair flow vector has at most M in{ D, A } distinct values. According to Lemmas 2 and 3, we can state that there are at most Min{ D, A } distinct sets D k. Furthermore, d D k we have the same λ [k], which proves the corollary. Let examine some other properties related to arc-connectivity on undirected networks. Thus, by the term network we mean an undirected network. Corollary 2 All first-level fair multi-flows in a capacitated k-arc connected network saturate in at least k arcs. At the end of the first step of the FMF algorithm, the multi-flow we obtain is necessarily saturated in at least one commodity d. Furthermore, the terminal nodes s d and t d cannot be connected through paths containing only non-saturated arcs because the flow value of the commodity d cannot be increased. Consequently, all disjoint paths (at least k) between s d and t d are saturated. Which is to say that at least k arcs are saturated at the end of the first step 4. Following the same idea, we can affirm that the terminal nodes of saturated commodities are placed in two distinct/complementary sets linked by saturated arcs. This defines thus a cut. At the end of the first step, let G 1 denote the subgraph obtained when removing the saturated arcs. Similarly, at the end of the k th step, let G k denote the subgraph obtained when removing from G k 1 the newly saturated arcs 5. At the end of the k th step, the terminal nodes of newly saturated commodities are necessarily disconnected in the subgraph G k and the number of components is strictly superior to this in G k 1. This can be shown by the fact that these new commodities were not saturated before running the latter step, and consequently the respective terminal nodes were in the same component. Thus, the latter component cannot remain any more connected in the new subgraph G k, and consequently the number of components becomes strictly superior to that obtained at the end of the preceding step. So after each step k, the number of components of G k is incremented by at least one. Since at the end of the first step the number of components is at least 2, it follows that after k 1 further steps the number of components becomes greater or equal to k + 1, which proves the following corollary: 4 A large number of commodities could be thus saturated during the first step. 5 According to Lemma 3, this is strictly equivalent to removing all saturated arcs from the initial network N. 10

11 Corollary 3 At the end of the k th step of the algorithm FMF, G k is composed of at least k + 1 components. The above corollary shows that each set of newly saturated links, disconnect some component of the subgraph constructed with previously nonsaturated links. This result suggests to apply the algorithm consecutively in subnetworks with respect to new components. This principle has already been used by Georgiadis et al. (see [9]), when studying lexicographically load-balanced networks. However their approach works in a limited scope, that is single-commodity networks and, as stated by authors, further investigations are needed to generalize it for multi-commodity networks. We think that the results and models presented here could make a rather good tool for achieving this goal. 4 Application to routing in telecommunication networks Here we shall study the fair routing problem in telecommunication networks. Generally speaking, the routing problem for telecommunication operators consists of two consecutive tasks: a good estimation of traffic demand, and the subsequent determination of the appropriate routes, or the routing of these multiple commodities in the underlying network. The latter problem is usually modeled as a multi-flow problem in a capacitated network with respect to a traffic demand matrix. Forecasts of traffic demand (summarized in the matrix) are generally expressed as amounts of traffic (e.g. in Mbs/sec) to be transported between two nodes. So long as other constraints (such as single routing etc.) are not considered, this problem can be modeled and solved as a simple linear programming one. In practice, routing has been seen as a component of other more complex and general problems related to the design and/or survivability of telecommunication networks ([2, 11]). The problem we shall consider in this section is obviously a routing problem with some particular constraints and with specific objectives. We aim to obtain the fairest distribution of resources via routing, that is, each demand has to be served fairly, at well as network resources will allow. This is especially suited for elastic traffic flows. Let have a look now at work devoted to fairness and fair routing. Some questions related to these issues, given extensive coverage in the literature, have arisen with world-wide Internet deployment. A key issue is studying how the congestion control mechanisms influence the resource sharing among the competing sessions, how fair is it and how integrating the fairness as design objective for such mechanisms, (see for example [12, 5, 16, 18] and the references therein). Some work has also been devoted to the static routing case, (the connections and the corresponding routing paths are given), where the source rates are subject to change. In [3], the author presents 11

12 the progressive filling algorithm for achieving a max-min fair distribution of resources to flows for the fixed single path routing case. Other algorithms for the static case, are given in [4, 6] etc. Furthermore, in [6], the authors propose an extension of such an algorithm, (it is a heuristic), for computing max-min fair routing. Note that the max-min fair routing studied in [6] is precisely the fair splittable multi-flow problem considered in this paper. Another research direction is investigating Semi-Definite Programming to express flow problems in the telecommunications context, considering several variants of fairness, (see [8]). Fairness finds also application in issues related to load balancing. Indeed, maximizing the minimum amount of resources distributed to source-sink commodities can be seen as minimizing the maximum load supported by links. Some relevant work on balanced networks is presented in [9] where the authors propose an approach for lexicographically optimal balanced networks. The problem considered is allocating bandwidth between two endpoints of a backbone network so that the network be equitably loaded, and therefore it is limited to the case of single-commodity network. In contrast, we consider a more general case here, that is to say, the multi-commodity network and some models presented here seems to be applicable to the load-balancing problems. In [19], we have also studied the problem of max-min fair routing, but in a particular context: routing of TCP/IP connections, (each of them considered as a separate commodity), which leads to some LP model quite different from these presented in this paper. This previous study was the starting point for the work presented here. Mathematical formulation. In addition to the notation given in section 2, let us provide some complementary notation necessary to model the problem: - The multi-commodity is represented by a given (elastic) traffic demand matrix: for each commodity we have a source s d, a sink t d and the associated traffic value T d. Therefore, the vector λ represents the flow ratio vector (to be maximized lexicographically) and not simply the flow vector as presented above in section 2. Indeed, as it can be seen below (see constraints (13 and 14)), the flow value associated to each demand is given by the product of its ratio and the given traffic value. - h(d) denotes the collection of all directed paths r from the source s d to the sink node t d of demand d. - fr d gives the flow value on path r for demand d. The notation (i, j) r means that arc (i, j) is contained in the path r. We have opted here for an arc-path (also known as path flow) formulation, which allows greater facility and flexibility when considering particular constraints such as path length or delay constraints. Telecommunication operators often wish to limit the length of routing paths. This requirement 12

13 can easily be addressed by this kind of formulation where routing paths are explicitly expressed. The fair routing problem can then be computed using the same algorithm (FMF) as given in section 2. The standard arc-path formulation of problem P k is given as follows: (i, j) A d D p:1 p<k d D \ ( 1 p<k D p ) Max r h(d),d D,(i,j) r r h(d) λ [k] f d r C ij (ω ij ) (12) f d r = λ[d] T d (π d ) (13) λ [k] T d r h(d) f d r 0 (π d ) (14) d D, r h(d) f d r 0 (15) where ω ij (respectively π d ) are the dual coefficients of constraints (12) (respectively constraints (13 and 14)). Inequality (12) indicates that the traffic value traversing the arc (i, j) does not exceed its capacity. In view of inequalities (13 and 14), the expression r h(d) f r d gives the amount of traffic with respect to demand d. The latter constraints are called traffic constraints. Finally, the constraints (15) indicate that path-flow values must be non-negative. A few words now about solving the multi-commodity flow problem P k. Obviously, we cannot make all candidate paths explicit, owing to their large number. We therefore use a set of initial paths and solve the problem as like a simple LP. However, the obtained solution is not necessarily optimal. We therefore use a column generation method: that is, we generate some new routing paths 6 through a path generator. We search for these paths in the initial graph valued with dual coefficient values ω ij. The reduced cost associated with each column (path r h(d)) is given by (i,j) r ω ij π d. Recall that according to the principle of Simplex algorithm, a candidate path for entering the basis and improving the current solution must have a positive reduced cost (for a maximization problem). Hence, for instance, we have to find d D, some path r h(d) maximizing (i,j) l ω ij + π d, which is strictly equivalent to minimizing (i,j) r ω ij or finding the shortest path from s d to t d. We iterate solving the modified LP program until we cannot generate any new columns, that is, we cannot find any path r such that (i,j) r ω ij < π d. Obviously, the solution obtained at this last step is an optimal one. Let us finally point out that in the general case we could use the Dijkstra shortest path algorithm. However if we have to limit the length of the routing paths, we use a modified version of the Ford-Bellman algorithm, which allows the search for shortest paths having at most e arcs. 6 Each column is represented here by a path. 13

14 Procedure Compute D k : In the same way as in the previous section we search for D k via the procedure Compute D k where the problem D q k can be stated mathematically as follows: d D \ ( (i, j) A d D p:1 p<k 1 p<k D p ), d q Max ɛ r h(d),d D,(i,j) r r h(d) r h(d) r h(q) f d r C ij (16) f d r = λ[d] T d (17) f d r λ [k] T d (18) f q r λ [k] T q + ɛ (19) d D, r h(d) f d r 0 (20) This problem is solved in a similar way to the preceding problem P k. Feasible routing versus fair routing. Let us now compare the feasible routing and fair routing problems. Assuming that the network resources are sufficient to route completely the traffic demands, it can be shown that each first-level fair multi-flow is feasible in this sense. Note firstly that by feasible routing we understand some multi-flow that satisfies all traffic requirements and capacity constraints, as stated below: (i, j) A fr d C ij d D r h(d),d D,(i,j) r r h(d) f d r T d d D, r h(d) f d r 0 With respect to the formulation of the first-level fair multi-flow problem (P 1 ), we observe that all constraints related to the feasibility routing problem are already expressed. In view of both formulations, we will necessarily obtain λ [1] 1 as solution for problem P 1, (if some feasible routing exists). Conversely, if λ [1] < 1 7, then the feasibility routing problem with respect to the given traffic demand matrix has no solution. We have thus shown that all fair routings are necessarily feasible routings when some feasible routing exists. 7 λ [1] value can be interpreted as routing feasibility index. 14

15 Network NET 11 NET 15 NET 26 NET 41 Nodes Arcs Demands Table 1: Description of test networks 5 Computational results Our algorithm for computing fair routing was implemented in C++ with CPLEX 8.0. All tests were run on a machine with the following configuration: Sun 4000, SOLARIS 2.5, 6 UltraSparc 167Mhz processors, 256 MB of RAM. We have implemented both models, arc-node and arc-path formulation, by using Simplex and interior-point methods. It should be noted that Simplex is an exponential-time method, widely used and well optimized (which is known to give much better results even than some polynomial-time algorithms e.g. Khachian s ellipsoid one). However, this method is not as performant as the interior-point method for large size problems. In Table 1, we summarize the main characteristics of four undirected network instances that we used for our computations. The second network, (NET 15 ), is a fully meshed network and the others are network instances dimensioned to route completely the traffic associated with the demand matrix. In the second table we report the algorithm s computational times (using Simplex for arc-path formulation) with respect to the allowed maximal length of routing paths. Notice that the elapsed time depends essentially on the size of the network and the number of traffic demands. Max Length NET 11 NET 15 NET 26 NET unbounded Table 2: Total computational time (in seconds). In table 3 we study the tradeoff between the two LP models described above (arc-node versus arc-path), in respect of size and CPU time. In our tests, we have used Simplex for implementing arc-path model and interiorpoint method for arc-node formulation model of fair routing. At a first glance, the arc-node formulation seems to be attractive (polynomial), but in practice (as shown in table 3) it becomes highly time-consuming, especially for not meshed networks (NET 26 and NET 41 ). For a network with N nodes, A arcs and D commodities, the arc-node formulation contains ( A + N D ) constraints (since it contains one capacity constraint per arc 15

16 and one mass balance constraint for each node and commodity combination), and about ( A D ) variables. In contrast, the arc-path formulation has a quite simple constraint structure and it contains only ( A + D ) constraints (in addition to the non-negativity restrictions imposed to path-flow values). However the number of path-variables will typically be enormous, (growing exponentially in the size of the network), but using the column generation method allows us not to make all of them explicit. Furthermore, in the light of linear programming theory we can state that at most A + D paths carry positive flow in some optimal solution to the problem. Thus, a large part of traffic demands is somehow guaranteed to be mono-routed with arc-path model. This result could be interesting in environments where the number of routes is restricted. NET 11 NET 15 NET 26 NET 41 A/P A/N A/P A/N A/P A/N A/P A/N CPU time (s) Nb. Rows Nb. Cols Table 3: Comparison of arc-node versus arc-path models. We think that is nevertheless interesting to compare fairness (as discussed in this paper) and throughput objectives. For this, we have firstly implemented the simply max-flow problem with respect to given commodities. It is obvious that these two objectives (throughput and fairness) are often contradictory and maximizing network throughput as primary objective may lead to gross unfairness: in the worst case some commodities may get a zero throughput. Providing both efficiency and some form of fairness is a key objective of congestion control in telecommunication networks. To this aim, we compute max-flow related to first-level fair routings (multi-flows). Some comparisons are presented in Table 4. The Fair Thruput column shows the global throughput of all connections routed across fair paths. The Max. Thruput column gives the maximum overall network throughput, and Max. Thruput 1 st level, the maximal flow value performed by all first-level fair routing. The divergence between the maximum throughput 1 st level and fair throughput is given in the column Gap 8 1. The column Gap 2 reports the divergence between the network throughput and the maximum throughput of 1 st level fair routing. In view of the obtained computational results, we can observe that fairness has a high cost in the throughput performances. This is even more pronounced especially for not meshed networks (e.g. NET 26 and NET 41 ). We can simply explain this by the fact that the routing paths in such networks are often longs, consuming thus precious resources. Let us point 8 The gap between a and b is computed as a b a 100%. 16

17 Network Fair Thruput Max. Thruput Max. Thruput 1 st level Gap 1 Gap 2 NET % 20.4% NET % 2.82% NET % 50.44% NET % 49.12% Table 4: Comparing fairness to network throughput. out that solutions maximizing the network throughput constantly neglect some costly traffic demands, which is not acceptable for telecommunication operators. However, as shown from the obtained results, the gap between fairness and throughput is smoothed if we limit ourselves to the first level of fairness. In some situations (i.e. fully or highly meshed networks, e.g. Net 15 ), this could be an acceptable compromise between fairness and throughput objectives. 6 Concluding remarks In this paper we have presented an iterative algorithm for computing fair multi-flows in a capacitated network. It is a linear programming based approach that permits a lexicographical maximization of the flow vector. We have also shown that our algorithm runs in polynomial time. We have also examined some theoretical properties and provided an optimality proof sketch of the proposed computation approach. An application case study in telecommunication area is also reported. It is worth noting that in this case the fair routing computing approach copes with a homogeneous traffic increase/decrease across all demands and it is mostly predestined to elastic traffic. The problem considered in this paper has been modeled both by an arc-node formulation for the general case and an arc-path formulation, better suited to telecommunications area. The latter model allows greater facility and flexibility in considering particular path constraints, such as bounded length or limiting in some level the number of obtained routing paths. This result could be especially interesting if applied to IP routing. Furthermore, as stated before, the results and models presented here could be a rather good starting point for studying load-balancing problems in networks. Another interesting research direction for the near future is studying the unsplittable case, (which is known to be NP-complete), where the theoretical results developed for the fair splittable multi-flow case are not applicable. 17

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