P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....packing and covering problems. Matula and Shahrokhi [11] were the rst who developed a combinatorial strongly polynomial approximation algorithm for the uniform concurrent ow problem. Their method was subsequently generalized and improved by Goldberg [4] Leighton et al. 10] Klein et al. [9], Plotkin et al. 13] and Radzik [14] A fast version that is particularly simple to analyze is due to Garg and K onemann [3] Independently, similar results were given by Grigoriadis and Khachiyan [6] In two recent papers Jansen and Zhang extended the latter approach and discussed an ....

....LM minprimal : minfminprimal; g, maxdual : maxfmaxdual; dualg if LG 100000 f LG : 100000 , for e 2 E : l(e) l(e) 100000 g g while minprimal (1 6 )maxdual cmax : for T 2 T i : c(T ) c(T ) j T i j 2. 3 Integral Solution Pure Combinatorial Approach Klein et al. [9] describe an approximation algorithm for the unit capacity concurrent ow problem that can be modi ed to approximate an integral solution to the minimum multicast congestion problem without the necessity to round. The algorithm is similar to the one presented in Section 2.2.1. However, instead of ....

P. Klein, S. Plotkin, C. Stein, E. Tardos, Faster Approximation Algorithms for the Unit Capacity Concurrent Flow Problem with Applications to Routing and Finding Sparse Cuts, SIAM J. on Computing 23 No.3, 466-487, 1994.

....clusters. In some applications, it is necessary, or advantageous, to have overlapping clusters, as discussed earlier. The problem of nding nonoverlapping clusters where the objective is to develop only two clusters with minimum number of inter cluster links [25, 19] or minimum ratio cut (see [35, 26, 34, 51]) has drawn considerable attention from researchers. In the VLSI domain this is known as the twoway partition problem. Kernighan et al. 25] and Fiduccia et al. 19] developed e ective heuristics for this problem, which are still extensively used. Clustering techniques have also been widely used ....

P. Klein, S. Plotkin, C. Stein and E. Tardos. Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts. SIAM Journal of Computing, 23, pages 466-487, 1994.

....maximum edge congestion by 2 . Our algorithms are approximation algorithms that find e optimal solutions, This chapter contains joint work with Tom Leighton, Fillia Makedon, Serge Plotkin, 15va Tardos and Spyros Tragoudas [42] and joint work with Philip Klein, Serge Plotkin and 15va Tardos [35]. ones in which A (1 d )A . In this chapter, we describe the first combinatorial approximation algorithms for the con current flow problem. Given any positive , the algorithms find an optimal solution. The running times of the algorithms depend polynomially on and are significantly ....

....integral solutions that are as strong as the results we have obtained about non integral solutions. For some cases This chapter contains joint work with Tom Leighton, Fillia Makedon, Serge Plotkin, lva Tardos and Spyros Tragoudas [42] and joint work with Philip Klein, Serge Plotkin and lva Tardos [35]. CHAPTER 3. APPLICATIONS OF MULTICOMMODITY FLOW of interest, however, we can obtain rather strong results. The ability to do so is interesting because the integer multicommodity flow problem seems to be harder than the non integral one: the integer problem is NP hard, while, as we have ....

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P. Klein, S. A. Plotkin, C. Stein, and 1. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Joural o Computig, 23(3):466 487, June 1994.

....possible. In section 6.1 we will analyze the ow scheduling problem in terms of ow graph problems, mainly variants of the multi commodity ow problem. In section 6. 2 we describe an approximation algorithm for the multi commodity ow problem proposed by Stein [22] and polished later by Klein [14] and Leighton [17] This algorithm is the basis of our ow scheduling algorithm, though we had to modify it because in our case the variables (commodities) may have more than one destination. The operand paths of those variables with more than one consumer are ideally mincost Steiner trees. 1 ....

....solution takes O(k 2 n 2 p km log (n 1 DU) time using a linear programming approach. Better running times can be achieved with combinatorial approximation methods. The foundations for these methods were laid by Shahrokhi and Matula [20] in the late eighties. Stein [22] Klein [14] and Leighton [17] have developed the method into a complete algorithm. While they concentrated on the problem of minimizing the congestion of the edges (also called the concurrent ow problem) the algorithm was later extended for the minimum cost multi commodity problem, 32 in which the ....

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P. Klein, S.A. Plotkin, C. Stein, and  E. Tardos. Faster Approximation Algorithms for the Unit Capacity Concurrent Flow Problem with Applications to Routing and Finding Sparse Cuts. SIAM Journal on Computing, 23(3):466-487, June 1994.

.... (V; E) introduced here is the same as the notion of minimum achievable forwarding index for networks restricted to G of [4] It is well known that the multicommodity ow problems are NP complete [6] Approximation algorithms for these problems with a guaranteed performance bound is presented in [8, 13]. In the following two subsections, we show that if the graph has a regular structure, the computation of the ow number of a graph is not necessarily all that dicult. We compute the ow number of a satellite constellation where each satellite (except the ones on the orbit boundary) has six ....

P. Klein, S. Plotkin, C. Stein, E. Tardos, \Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts", SIAM Journal of Computing, vol. 23, no. 3, pp. 466-487, 1994.

....In 1990, Shahrokhi and Matula proved polynomial time convergence rates for a Lagrangianrelaxation algorithm for multicommodity flow. This caught the attention of the theoretical computer science research community, which has since produced a large body of research on the subject. Klein et al. [15] and Leighton et al. 19] and many others) gave additional multicommodity flow results. Plotkin, Shmoys, and Tardos [21] and Grigoriadis and Khachiyan [10, 11, 8, 9] adapted the techniques to the general class of packing covering problems, including mixed packing and covering problems. These ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J. Comput., 23(3):466--487, June 1994.

....shows no trace of this relation. The explicit connection with multicommodity flows was first made by Shahrokhi and Szekely [21] 22] motivated by the earlier work of Shahrokhi and Matula on the concurrent flow problem [20] For more recent work on multicommodity flows see [2] 18] 9] and [15]. 2. Background and Basic Notations. Our graph theoretical terminologies defined here are consistent with [5] Throughout our paper, we define all concepts relevant to our work. For concepts in topological graph theory in more detail, see [28] and [27] For g # 0, let S g and N g denote the ....

Klein, P., Plotkin, S., Stein, C., and Tardos, E., Faster approximation algorithms for unit capacity concurrent flow problems with applications to routing and sparsest cuts, SIAM J. Comput., 3(23) (1994), 466--488.

....as we know. 2 In the recent years, considerable research has gone into the design of pseudo polynomial time approximation algorithms for multicommodity flow feasibility problems, e.g. see Leighton, Makedon, Plotkin, Stein, Tardos, and Tragoudas [1991] Plotkin, Shmoys, and Tardos [1991] and Klein, Plotkin, Stein, and Tardos [1994]. We have investigated these approaches and did not see how they could substantially help solving the optimization problems that we investigate here. In particular, the results reported in Leong, Shor, and Stein [1993] and Borger, Kang, and Klein [1993] on rather small problem instances do not ....

Klein, P., Plotkin, S., Stein, C., and Tardos, ' E. (1994). Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts.

.... of such sparse, near optimal strategies was shown probabilistically [2, 13] our existence proof of the approximate solution for generalized packing is a generalization of the proof in [13] 2 Related Work Plotkin, Shmoys, and Tardos [16] generalizing a series of works on multicommodity flow [19, 11, 12]) gave approximation algorithms for general packing and covering problems similar to those we consider. For these abstract problems, their results are comparable to those in this paper, but for many problems their results are stronger. Most importantly, they give techniques for reducing the ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

....to use randomized rounding [5] 1.2 Previous results The problem defined in the previous section is related to multicommodity flow problems. Problems of this kind can be solved through linear programming in polynomial time (see e.g. 6] There are also a number of fast approximation algorithms [7, 8, 9, 10] for the multicommodity flow problem. In [11] Garg and Konemann present an interesting new approach that finds good approximate solutions to multicommodity flow problems solely through an iterative use of shortest path calculations. We decided to compare this algorithm with our algorithm. However, ....

P. Klein, S.A. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity flow problem with applications to routing and finding sparse cuts. In Proc. of the 22nd Ann. ACM Symp. on Theory of Computing, 1990.

.... combinatorial method which was originally proposed in the late eighties by Shahrokhi and Matula [1990] and has undergone since then many re nements, improvements and extensions [Garg and K onemann 1998; Goldberg 1992; Grigoriadis and Khachiyan 1996; Kamath et al. 1995; Karger and Plotkin 1995; Klein et al. 1994; Leighton et al. 1995; Radzik 1997; Plotkin et al. 1995; Villavicencio and Grigoriadis 1995] In this paper we refer to this method using acronym CMCF Combinatorial Multi Commodity Flow. The CMCF method is an iterative method based on minimization of a potential function which depends ....

Klein, P., Plotkin, S., Stein, C., and Tardos, E. 1994. Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts. SIAM J. Computing 23, 466-487.

....within a factor of (1 #) of optimal in time bounded by a polynomial in the size of the graph and 1 #. The key to analyzing the running time of this algorithm is an exponential potential function, which has been the basis for several subsequent papers as well. Klein, Plotkin, Stein, Tardos [KPST94] and Leighton, Makedon, Plotkin, Stein, Tardos, Tragoudas [LMP 95] subsequently improved and extended this result to derive an algorithm for the maximum concurrent flow problem for which the running time for a k commodity problem is competitive with the running time for k singlecommodity ....

....for each f # Q, the total flow on each edge e can be at most kc(e) Furthermore, the required subroutine to optimize over Q is still relatively easy: it requires k (single commodity) maximum flow computations. By incorporating randomization as first proposed by Klein, Plotkin, Stein, Tardos [KPST94] the effort for this step can be reduced to, in effect, a single max flow computation. This algorithm for the maximum concurrent flow problem is due to Leighton, Makedon, Plotkin, Stein, Tardos, Tragoudas [LMP 95] Finally, the dual variables maintained by this algorithm can be easily ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J. on Computing, 23:310--321, 1994.

....Therefore how to approximately solve these semidefinite relaxations efficiently, in both time and space, is practically important. All the previous work resorts to various general interior point methods [5, 113, 150] for this step. In the thesis we adapt the technique of Lagrangean relaxation [96, 105, 121, 87, 158] to cope with this crucial step. Specifically we use the framework of Plotkin, Shmoys, and Tardos [121] to obtain near optimal iv solutions to the semidefinite relaxations of MAXCUT and COLORING. Our results significantly reduce the work space required by the best known approximation algorithms ....

....they can handle instances that previous algorithms cannot even touch. The basis of our algorithms is the framework of Plotkin, Shmoys, and Tardos [121] which is related to the technique of Lagrangean relaxation. Lagrangean relaxation useful in exploiting the structure of mathematical programs [96, 105, 121, 87, 158]. It basically works as follows: Suppose we are given a set of constraints, over which an objective function is supposed to be optimized. Instead of solving it using general purpose algorithms such as ellipsoid methods [92, 68] or interior point methods [89, 147, 50] we could separate the ....

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P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466-- 487, June 1994. The preliminary version is [97].

....the maximum concurrent flow problem with uniform arc capacities. They introduced a length function which is exponential in the total flow going through that arc. They iteratively route flow along shortest paths with respect to the exponential length function. The method was refined by Klein et al. [22] and extended to handle arbitrary arc capacities by Leighton et al. 23] Plotkin, Shmoys, and Tardos [26] and Grigoriadis and Khachiyan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster randomized version; Radzik [27] ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

....pair of vertices to minimize the congestion. The (fractional) uniform flow problem is known to be solvable in polynomial time [17] and starting from the work of Shahrokhi and Matula [17] there have been a series of papers on how to approximately solve this problem faster [1] 11] 6] and [10]. The integral version is known to be NP hard [3, 16] There is a need for estimating the value of the congestion, since many important graph theoretical parameters are related to the congestion. For instance the congestion of a uniform flow provides for lower bounds for the bisection width [13, ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos, Faster approximation algorithms for unit capacity concurrent flow problems with applications to routing and sparsest cuts, SIAM Journal on Computing 3:23 (1994), 466--488.

....the maximum concurrent flow problem with uniform arc capacities. They introduced a length function which is exponential in the total flow going through that arc. They iteratively route flow along shortest paths with respect to the exponential length function. The method was refined by Klein et al. [22] and extended to handle arbitrary arc capacities by Leighton et al. 23] Plotkin, Shmoys, and Tardos [26] and Grigoriadis and Khachiyan [17] extended the method further to solve more general fractional packing and covering problems. Goldberg [13] proposed a faster randomized version; Radzik [27] ....

P. Klein, S. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

....Shahrokhi and Matula [19] give the first polynomial time, combinatorial algorithm for approximating the maximum concurrent flow problem with uniform capacities, and introduced the use of an exponential length function to model the congestion of flow on an edge. Klein, Plotkin, Stein, Tardos [12] improve the complexity of this algorithm using randomization. Leighton et al. 13] extend [12] to handle graphs with arbitrary capacities, and give improved run times when capacities are uniform. None of these papers considers the versions with costs. Grigoriadis and Khachiyan [8] describe ....

....the maximum concurrent flow problem with uniform capacities, and introduced the use of an exponential length function to model the congestion of flow on an edge. Klein, Plotkin, Stein, Tardos [12] improve the complexity of this algorithm using randomization. Leighton et al. 13] extend [12] to handle graphs with arbitrary capacities, and give improved run times when capacities are uniform. None of these papers considers the versions with costs. Grigoriadis and Khachiyan [8] describe approximation schemes for block angular linear programs that generalize the uniform capacity maximum ....

P. Klein, S. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

....better than LS2 does. Mao and Simha left open the upper bound of LS3 and the question of whether there exist better greedy algorithms. Aspnes, et al. 1, 2] designed an elegant, asymptotically optimal online algorithm, based on techniques to approximately solve offline multicommodity flow problems [25, 21, 20], which assigns to each request a shortest path with respect to the following exponential cost function: cost e (j) a j (e) l j c(e) Gamma a j (e) where a = 1 fl for any 0 fl 1. This algorithm, which we will call Exp Route, is O(log n) competitive, where n = jV j, on arbitrary ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

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P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

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P. Klein, S. A. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

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P. Klein, S. A. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

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P. Klein, S. A. Plotkin, C. Stein, and E. Tardos, Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts, SIAM J. Comput., 23 (1994), pp. 466-487.

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P. Klein, S. A. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

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Philip Klein, Serge Plotkin, Clifford Stein, and ' Eva Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

....are increased by a (1 ffl) factor, or to provide a proof that there is no feasible solution to the original problem. We also describe faster approximation algorithms for multicommodity flow problems with a special structure, such as those that arise in the sparsest cut problems studied in [8, 10, 9], and the uniform concurrent flow problems studied in [12, 9] if k m. 1 Introduction The multicommodity flow problem involves simultaneously shipping several different commodities from their respective sources to their sinks in a single network so that the total amount of flow going through ....

....there is no feasible solution to the original problem. We also describe faster approximation algorithms for multicommodity flow problems with a special structure, such as those that arise in the sparsest cut problems studied in [8, 10, 9] and the uniform concurrent flow problems studied in [12, 9] if k m. 1 Introduction The multicommodity flow problem involves simultaneously shipping several different commodities from their respective sources to their sinks in a single network so that the total amount of flow going through each edge is no more than its capacity. Associated with each ....

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P. Klein, S. A. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. Technical Report 961, School of Operations Research and Industrial Engineering, Cornell University, 1991.

.... linear programming) 4] Benders decomposition [3] the Lagrangean relaxation method developed by Held and Karp and applied to obtaining lower bounds for the traveling salesman problem [9, 10] the multicommodity ow approximation algorithms of Shahrokhi and Matula [18] of Klein et al. [13], and of Leighton et al. 15] the covering and packing approximation algorithms of Plotkin, Shmoys, and Tardos [16] and the approximation algorithms of Grigoriadis and Khachiyan [8] for block angular convex programs, and many subsequent works (e.g. 20, 6] In a later section we discuss some ....

....Packing and Covering The third line of research, unlike the rst two, provided guaranteed convergence rates. Shahrokhi and Matula [18] gave an approximation algorithm for a special case of multicommodity ow. Their algorithm was improved and generalized by Klein, Plotkin, Stein, and Tardos [13], Leighton et al. 15] and others. Plotkin, Shmoys, and Tardos [16] noticed that the technique could be generalized to apply to the problem of nding an element of the set fx : Ax b; x 2 Pg (9) where P is a convex set and A is a matrix such that Ax 0 for every x 2 P . In particular, as ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts. SIAM Journal on Computing, 23(3):466-487, June 1994.

....on interior point techniques, is due to Kamath and Palmon [6] and is slower by a factor of Omega Gamma k 1:5 p m=n) for constant ffl. Combinatorial approximation algorithms for various variants of multicommodity flow can be divided according to whether they are based on relaxing the capacity [14, 9, 10, 12] or conservation constraints [1, 2] In particular, the algorithm in [12] is based on relaxing the capacity and budget constraints. In other words, the algorithm starts with a flow that satisfies the demands but does not satisfy either the capacity or budget constraints. It repeatedly reroutes ....

P. Klein, S. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, June 1994.

....an associated cost and the goal is to find a flow of minimum cost that satisfies all the demands. Multicommodity flow arises naturally in many contexts, including virtual circuit routing in communication networks, VLSI layout, scheduling, and transportation, and hence has been studied extensively [7, 10, 14, 17, 12, 13, 18, 2, 16]. Since multicommodity flow algorithms based on general interior point methods for linear programming are slow [10, 19, 8] recent emphasis was on designing fast combinatorial algorithms that relied on problem structure. One successful approach has been to develop approximation algorithms. If ....

....and Plotkin, 9] who gave an O(ffl Gamma4 kmn 2 ) algorithm, where n is the number of nodes. This is more than n times slower than Radzik s deterministic algorithm [16] for the no cost version of the problem. Even better running times were achieved for special cases of the nocost problem [12]. It is interesting to note that adding costs does not significantly affect the running time of the interior point based algorithms [8, 19] The main contribution of this paper is a deterministic minimum cost multicommodity flow algorithm that runs in O(ffl Gamma3 kmn) time, essentially ....

P. Klein, S. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, June 1994.

....be used to convert the randomized algorithm into a deterministic algorithm. Often, there are faster algorithms that obtain comparable performance guarantees without solving the linear program. For instance, this is the case for packing and covering problems abstracting multicommodity flow problems (Klein et al. 1994; Goldberg, 1991; Radzik, 1995; Grigoriadis and Khachiyan, 1995; Plotkin et al. 1991; Karger and Plotkin, 1995) These algorithms are also faster than general linear programming for approximately solving fractional (relaxed) packing and covering problems. Such algorithms can be understood within ....

Klein, P., Plotkin, S., Stein, C., and Tardos, ' E. (1994). Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487.

....its local data and sending the results to all of its neighbors. Our algorithm is local in a sense that it does not depend on any global computations like shortest paths or min cost flow, that were the main subroutines of the previously known efficient combinatorial multi commodity flow algorithms [47, 49, 67]. The only previously known local approximate concurrent flow algorithm, due to Awerbuch and Leighton [8, 9] does not address the min cost problem. The main advantages of a local algorithm is ease of implementation in a distributed system where centralized control is either infeasible or very ....

....the goal of an (ffl; ffi) approximation algorithm is to find a flow of cost at most (1 ffi )B that satisfies (1 Gamma ffl) fraction of each demand. Approximation algorithms for various variants of multi commodity flow can be divided according to whether they are based on relaxing the capacity [70, 47, 49, 67] or conservation constraints [8, 9] In particular, the algorithm in [67] is based on relaxing the capacity and budget constraints. In other words, the algorithm starts with a flow that satisfies the demands but does not satisfy either the capacity or budget constraints. It repeatedly reroutes ....

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P. Klein, S. A. Plotkin, C. Stein, and ' E. Tardos. Faster Approximation Algorithms for the Unit Capacity Concurrent Flow Problem with Applications to Routings and Finding Sparse Cuts. Technical Report 961, School of ORIE, Cornell University, 1991.

....Awerbuch et al. 1994a; 1994b] and Kleinberg and Tardos [1995] Our routing and scheduling algorithms assume a central scheduler that makes all the decisions. In Awerbuch and Azar [1994] the techniques of this paper were extended to the case where 1 See, for example, Shahrokhi and Matula [1990] Klein et al. 1994], Leighton et al. 1995] Plotkin et al. 1995] and Karger and Plotkin [1995] 490 J. ASPNES et al. there are concurrent requests that have to be satisfied in a decentralized fashion. See Plotkin [1995] for a survey of different on line routing strategies. We note that another measure of ....

KLEIN, P., PLOTKIN, S., STEIN, C., AND TARDOS,E . 1994. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J. Comput. 23, 3, 466 -- 487.

....time. If this is not the case, our algorithm may still be useful if many shortest path computations are performed on the same graph, for in this case it may be worthwhile to precompute the decomposition. This is the case in various algorithms, e.g. approximate multicommodity flow computations. [KPS, PST]. The algorithm for arbitrary lengths first applies the shortest path algorithm due to Lipton, Rose, and Tarjan [LRT] to each region, obtaining shortest path distances between each pair of boundary nodes of the region. For each region, the algorithm constructs a complete directed graph on the ....

P. Klein, S. Plotkin, C. Stein & ' E. Tardos, "Faster Approximation Algorithms for the Unit Capacity Concurrent-Flow Problem With Applications to Routing and Finding Sparse Cuts," SIAM Journal on Computing 23 (1994), 466--487. 30

....in the work of Shahrokhi and Matula [22] on approximate solution of a multicommodity flow problem. Shahrokhi and Matula proved a polynomialbut rather high bound on the running time of their algorithm. A much faster algorithm for this problem was developed by Klein, Plotkin, Stein, and Tardos [10]. One important ingredient in the improvement is a formulation of approximate optimality (relaxed complementary slackness conditions) that works well in this setting. This multicommodity flow algorithm was generalized by Leighton, Makedon, Plotkin, Stein, Tardos, and Tragoudas [14] Plotkin, ....

P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23(3):466--487, June 1994.

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P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

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P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

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P. Klein, S. Plotkin, C. Stein, and  E. Tardos. Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts. SIAM Journal on Computing, 23:466-487, 1994.

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P. Klein, S. Plotkin, C. Stein and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. Proc. 22nd Ann. ACM Symp. on Theory of Computing, 310-321, 1990.

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P. Klein, S. Plotkin, C. Stein, and ' E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM Journal on Computing, 23:466--487, 1994.

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P. Klein, S. Plotkin, C. Stein and E. Tardos, Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts, Proc. 22nd Ann. ACM Symp. on Theory of Computing (1990), 310 - 321.

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P. Klein, S. Plotkin, C. Stein and E. Tardos, Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts, Proc. 22nd Ann. ACM Symp. on Theory of Computing (1990), 310 -- 321.

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P. Klein, S. Plotkin, C. Stein and E. Tardos. Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts. SIAM Journal of Computing, 23, pages 466-487, 1994.

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P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J. Comput., 23(3):466--487, 1994.

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P. Klein, S. Plotkin, C. Stein and E. Tardos, Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts, SIAM Journal on Computing, 23 (1994), 466-487.

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P. Klein, S. Plotkin, C. Stein, and E. Tardos. Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J. Comput., 23(3):466--487, 1994.

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P. Klein, S. Plotkin, C. Stein, and  E. Tardos. Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts. SIAM Journal on Computing, 23:466-487, 1994.

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P. Klein, S. Plotkin, C. Stein, E. Tardos, Faster approximation algorithms for unit capacity concurrent flow problems with applications to routing and sparsest cuts, SIAM J. Computing, 3(23) (1994), pp. 466-488.

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P. Klein, S. Plotkin, C. Stein, E. Tardos, Faster approximation algorithms for unit capacity concurrent ow problems with applications to routing and sparsest cuts, SIAM J. Computing, 3(23) (1994), pp. 466 488.

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P. Klein, S. Plotkin, C. Stein and E. Tardos, Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts, Proc. 22nd Ann. ACM Symp. on Theory of Computing (1990), 310 - 321.

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P. Klein, S. Plotkin, C. Stein, E. Tardos, \Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts", SIAM Journal of Computing, vol. 23, no. 3, pp. 466-487, 1994.

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P. Klein, S. Plotkin, C. Stein and E. Tardos, Faster approximation algorithms for the unit capacity concurrent ow problem with applications to routing and nding sparse cuts, Proc. 22nd Ann. ACM Symp. on Theory of Computing (1990), 310 - 321. 10

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