Some results on max-min fair routing Dritan Nace, Linh Nhat Doan University of Technology of Compiegne. Laboratory Heudiasyc UMR CNRS 6599, 60205 Compiègne Cedex, France. Phone: 00 33 344234302, fax: 00 33 344234477 Email: nace@utc.fr, doannhat@hds.utc.fr Abstract Variants of the flow problem in a graph with edge capacities frequently arise in operations research and among them, network flow problems have been already extensively studied. We study here the max-min fair routing which is expressed through multi-commodity flows. This problem is equivalent to the lexicographical maximization of the flow vector in capacitated multi-commodity networks. We resolve this problem through an iterative algorithm and show the polynomiality of this approach. It is based on a new linear programming model that allows lexicographical maximization of the vector of individual flows in multi-commodity networks. Some theoretical analyses in addition to those presented in our previous works are also given. Keywords: Routing, Max-min Fairness, Graph Theory, IP Networks. 1 Introduction Flow problems in networks have been extensively studied since the seminal work of Ford and Fulkerson in this area ([6]). A number of variants of the flow problem in a graph with edge capacities frequently arise in operations research and other fields (see [1]). We are interested here by distributing the flow (max-min) fairly. This latter problem, somehow similar to balanced (multi-commodity) flow problem (see [1]), 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 the so-called general max-min fairness. In other words, we consider here the lexicographical maximization of the vector of individual flows in multi-commodity networks as applied to max-min fair routing problem. Related work: Number of studies related to max-min fairness have been presented and analyzed in [1, 3, 5, 9, 12, 13] and the references therein. The first problem studied in the literature is the max-min fair flow one in a single-source multi-sink 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 max-min 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 [12], the author has put forward an This work is supported by France Telecom R&D.
elegant method for the max-min fair single-source splittable flow problem. Conversely, the max-min fair singlesource unsplittable flow problem is shown to be NP-complete: special cases of the latter problem include several fundamental load-balancing problems, (see [9]). 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 one, 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-commodity flows with respect to given commodities. We call this problem the max-min fair multi-commodity flow problem and we will formally define it in the next section. Notice that issues on max-min fairness are a key concern in numerous studies inspired from applications in the telecommunications area such as congestion and flow control, load-balancing or max-min fair routing. Let have a look now at some work devoted to max-min fairness and max-min 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 (max-min) fairness as design objective for such mechanisms, (see for example [4, 8, 11] and the references therein). Some work has also been devoted to the static routing case (the connections and the corresponding routing paths are given, see [3, 5]), where the source rates are subject to change. In [3], the author presents the progressive or water filling algorithm for achieving a max-min fair distribution of resources to flows for the fixed single path routing case. This can also be adapted to the fixed multi-path routing problem but it is not at all obvious when the routing paths are not given and have to be found. In [5], the authors propose a LP model for the static case and in addition an extension of such an algorithm, (hence not optimal), for computing variants of fair routing. Recently in [15], the authors present an alternate algorithm for achieving max-min fair routing in the general case. Max-min fairness finds also application in issues related to load balancing. Some relevant work on balanced networks is presented in [7] 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 the models presented here are applicable to this load-balancing problem. Finally in some recent works ([16, 17]), are proposed general algorithms for max-min fairness. However we intend to provide a deeper study on a particular problem that is max-min fair multi-commodity flow. Contribution of this work: In this paper we introduce a polynomial-time algorithm for the max-min fair multi-commodity 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 max-min fair multi-commodity flows. This work, however, is limited to the splittable flow case. We recall that in [13], we have also studied the problem of max-min fair routing and proposed a quite complicated LP model. In this paper, we present a much more simple LP model which allows us to prove the polynomiality and extend the theoretical results presented before by showing that the property of saturating links in a unique order holds even for commodities. This previous study was the starting point for the work presented here. The paper is organized as follows: in section 2 we provide a mathematical framework and introduce an iterative (polynomial) algorithm for achieving an optimally max-min fair splittable multi-commodity flow in a capacitated network. Next, we examine the theoretical properties and also provide proof of the optimality of our approach. Finally, in section 3 we give some concluding remarks. 2 Mathematical modeling and resolution approach In this section we introduce an approach for the fair multi-commodity flow problem. definitions and notation useful for the remainder of the paper. First, let us give some
2.1 Definitions and notation A capacitated network is a given triplet N = (V, A, C), where V is a nonempty finite set of nodes. A is the set of arcs, and C ij is the capacity associated with the arc (i, j). 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 λ[d] 0), in network N with respect to commodity d is a function from A to R + such that: i V, fij d λ[d], i = s d fji d = λ[d], i = t d (1) j:(i,j) A j:(j,i) A 0, otherwise where f d ij give the arc-flow values. A multi-commodity flow f is composed of flows f d with respect to commodities d D such that: (i, j) A, d D f d ij C ij (2) This multi-commodity flow is called feasible. Definition 1 Given a multi-commodity 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-commodity 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. Notice that the coordinates of the vector λ are the decision variables of our problem. Definition 2 A vector λ 1 is called lexicographically greater (respectively smaller) to a second vector λ 2 if there exists k such that i < k, λ 1[i] = λ 2[i] and λ 1[k] > (respectively <) λ 2[k]. We denote this by λ 1 λ 2 (resp. λ 1 λ 2). - Let F denote the set of all the feasible multi-commodity flows in a given capacitated network N with respect to a set of commodities D. Definition 3 A multi-commodity flow f, whose flow vector is λ, is called max-min 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 a max-min fair multi-commodity flow as max-min fair. An important property of an optimally max-min fair multi-commodity 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, 10]). Mathematically, this can be stated as follows: Given a flow vector λ, where λ[d] gives the value of flow d, we call λ feasible if there is a feasible multi-commodity flow satisfying it. Then, the flow vector λ is said max-min fair if it is feasible and for any other feasible flow vector λ, such that d D : λ [d] > λ[d], than p D : λ [p] < λ[p] λ[d]. This alternate definition is derived from the max-min fairness one 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 definitions can be easily generalized by using weights. 2.2 Resolution approach We introduce here an algorithm for resolving the fair multi-commodity 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-commodity 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. 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 MFMF, (Max-min Fair Multi-Commodity Flow) Given: A capacitated network N(V, A, C) and a set of commodities d D. Required: Find the fair multi-commodity 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), compute the current λ [k] value and the set D k ); - p [k, k + D k 1], 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-commodity flow is thus max-min fair. where the problem P k is 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 : The problem P k can be modeled as a multi-commodity flow problem aiming to achieve some level of fairness. We have modeled this problem through a classic arc-node LP formulation as follows: Max λ [k] (i, j) A, d D f d ij C ij (3) i V, d D p:1 p<k, fij d fji d = j:(i,j) A j:(j,i) A i V, d D \ ( D p), fij d 1 p<k j:(i,j) A j:(j,i) A f d ji λ[d], i = s d λ[d], i = t d 0, otherwise λ [k], i = t d λ [k], i = s d = 0, otherwise (4) (5) (i, j) A, d D, fij d 0 (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. At the end of the solving procedure we obtain a multi-commodity flow, λ [k] and the dual values associated with constraints. 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 accidentally saturated. According to some fundamental results of linear programming theory (complementary slackness property), the constraints satisfied by the solution with equality are these whose dual coefficient values are strictly positives (for the standard LP formulation). So, for determining the set D k, we just need to have a look at some dual coefficients values (let us call them π(s d, d), π(t d, d)) associated with constraints (5). 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]). 2.3 Some theoretical properties We discuss here the optimality and polynomiality of our approach, and we highlight some properties of max-min fair multi-commodity flows and the max-min fair flow vector. We recall that the existence and uniqueness of the max-min fair allocation vector (similar to flow vector) for the fixed single path flow case is already given in [3, 10]. In the following we extend these results for the general multi-path flow case. The starting point for this is showing how to build some max-min fair multi-commodity flow. Thus, the primary objective is to prove that the MFMF
algorithm given above enables the max-min fair multi-commodity flow computing in polynomial time for a given capacitated network and set of commodities. Let us begin by considering the correctness of our iterative algorithm, which is not obvious. Indeed, distinct multi-commodity 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 MFMF algorithm we get some multi-commodity flow f and the k th coordinate value of the vector λ. 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 max-min fair) of the multi-commodity flow computed by our FMF algorithm. For this, we need to state some preliminary result: Lemma 1 All multi-commodity flows obtained at the end of the MFMF Algorithm saturate commodities in a given unique order. Proof: The following proof is analogous of this given in [13] concerning the order of saturating arcs. Let us consider the set of multi-commodity flows satisfying all constraints associated with the problem P k, (step k). Obviously, for each multi-commodity 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-commodity flows obtained as solution of problem P k. D k is not empty. Indeed, without loss of generality, let us suppose by absurdity that there exist two multi-commodity 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-commodity flows are such that Q 1 D \ ( 1 p<k Dp) (resp. Q2 D \ ( 1 p<k Dp) represents the set of newly saturated commodity flows for Φ1 (resp. Φ 2) and Q 1 Q 2 =. Let us set Φ = (Φ 1+Φ 2 ). It is obvious that multi-commodity flow Φ satisfies capacity 2 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 Dp). In these conditions, we increase at the same pace the flow value to each f d ij with respect to non-saturated commodities (d D \( 1 p<k Dp)) 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. Furthermore, in the view of the FMF algorithm, it can easily be seen that commodities are saturated in an unique order fixed through the consecutive steps of the algorithm. Notice also that the commodities contained in the same D k, are interchangeable in the saturating order since they saturate simultaneously at the same value λ [k]. Now, the optimality and the polynomiality of the algorithm can be deduced as follows: Theorem 1 The multi-commodity flow obtained at the end of the MFMF algorithm is a max-min fair one and it can be obtained in polynomial time. Proof: 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-commodity flow (resp. multi-commodity flow obtained by FMF), and γ (resp. λ) the corresponding flow vector. As noted before, γ and λ give the respective flow vector (of γ and λ) reordered in increasing order of magnitudes. Let us suppose by absurdity that γ γ, that is to say 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-commodity flow Γ satisfies all constraints of the problem P l for every l [1, k 1]. According to the above lemma we can affirm 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 first part of the theorem. Concerning the polynomiality of the approach it can be deduced from the fact that the algorithms run in at most D steps and each step is intended to resolve one LP problem. Then, the polynomiality follows from the fact that each LP problem is stated as a classic multi-commodity flow problem which is known to be polynomial-time, (Khachian s ellipsoid algorithm, Karmarkar s interior-point algorithm). 2.4 Max-min fair routing Here we shall briefly discuss the max-min fair routing problem in telecommunication networks. Generally routing problems are usually modeled as a multi-commodity flow problems in a capacitated network with respect to a
traffic demand matrix. Forecasts of traffic demand (summarized in the matrix T where T d gives the target value associated with commodity d) are generally expressed as amounts of traffic (e.g. in Mbs/sec) to be transported between two nodes. We aim to obtain the (max-min) fairest distribution of resources via routing, that is, each demand has to be served fairly, at well as network resources will allow. We just need now to slightly modify the model presented above by introducing for each commodity some 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. Indeed, in constraints (4) and (5) in the model described before, the flow value associated to each demand should be replaced by the product of its ratio and the given traffic value, that is λ[d] T d. 3 Concluding remarks In this paper we have presented an iterative algorithm for computing max-min fair multi-commodity 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. References [1] R.K. Ahuja, T.L. Magnanti and B. Orlin Network Flows : Theory, Algorithms and Applications. Prentice Hall, 1993. [2] R.K. Ahuja Balanced linear programming problem European Journal of Operational Research, 101, 1, pp. 29-38, Aug, 1997, [3] A. Bertsekas and R. Gallager. Data Networks. Prentice-Hall, Engelwood Cliffs, N.J., 1992. [4] D. Chiu and R. Jain. Analysis of the increase and decrease algorithms for congestion avoidance in computer networks. Computer Networks and ISDN Systems, 17:1 14, 1989. [5] G. Fodor, G. Malicsko, M. Pioro and T. Szymanski. Path Optimization for Elastic Traffic under fairness Constraints. In Proc. of ITC 2001. [6] L.R. Ford and D.R. Fulkerson. Flows in Networks. Princeton University Press, N.J., 1962. [7] L. Georgiadis, P. Georgatsos, K. Floros, S. Sartzetakis Lexicographically Optimal Balanced Networks. In Proc. Infocom 01, pp. 689-698, 2001. [8] F. P. Kelly, A. Maulloo, and D. Tan. Rate control in communication networks: shadow prices, proportional fairness and stability. Journal of the Operational Research Society, vol. 49, pp. 237-252, 1998. [9] J. Kleinberg, Y. Rabani, and E. Tardos. Fairness in routing and load balancing. In Proc. of the 35th Annual Symposium on Foundations of Computer Science, 1999. [10] J.-Y. Le Boudec Rate adaptation, congestion control and max-min fairness: a tutorial. http://ica1www.epfl.ch/4ps files/leb3132.pdf, December 2000. [11] L. Massoulié and J.W. Roberts. Bandwidth sharing: objectives and algorithms. In Proc. of IEEE INFO- COM 99, 1999. [12] M. Megiddo. Optimal flows in networks with sources and sinks. Mathematical Programming, 7, 1974. [13] D. Nace. A linear programming based approach for computing optimal splittable fair routing. In Proc. of The Seventh IEEE Symposium on Computers and Communications 2002, ISCC 2002. pp. 468-474, July 2002, Taormine, Italy. [14] D. Nace, L.N. Doan. A polynomial approach to the fair multi-commodity flow problem. Technical Report, Heudiasyc, UTC, November 2002.
[15] M. Pioro, P. Nilsson, E. Kubilinskas and G. Fodor. On efficient Max-min Fair Routing ALgorithms. Proc. of ISCC 2003. [16] B. Radunovic, J.-Y. Le Boudec. A Unified Framework for Max-Min and Min-Max Fairness with Applications. Technical Report IC-200248, EPFL, July 2002, http://lcawww.epfl.ch/publications/radunovic/tr02 048.ps. [17] A. Tomaszewski. A polynomial algorithm for solving a general max-min fairness problem. In the Proceedings of the second Polish-German Teletraffic Symposium, PGTS 2002, Gdansk, 2002.