N. Garg, V. Vazirani, M. Yannakakis, "Approximate max-flow min-(multi)cut theorems and their applications," Proc. 25th ACM Symp. on Theory of Computing, 1993, pp. 698--707. 182  Home/Search   Document Details and Download   Summary   Related Articles   Check

....an optimal solution can be computed in this case in polynomial time by using the ellipsoid method. In fact, interior point algorithms can also achieve polynomial time since it is possible to decrease the number of constraints to be polynomial by reducing the fvs problem to multi commodity flow [2]. But, then the linear program is not positive anymore. Using general linear programming methods is usually undesirable, and algorithms that exploit the combinatorial structure of the problem are sought when possible. Since we deal with approximation algorithms, we can actually settle for an ....

....pairs. This problem is also NP Hard: as demonstrated later, we can reduce the feedback problems to the multi cut problem in directed networks. Recently, there has been much progress in approximating the multi cut problem in the undirected case. Most notably, Garg, Vazirani and Yannakakis [2] achieved a O(logk) approximation factor for this problem. They introduced a novel sphere growing technique, refining and simplifying previous works of [11, 7] Actually, the O(log k) factor achieved in [2] is with respect to the optimal fractional solution of the multi cut problem. Garg, ....

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N. Garg, V.V. Vazirani and M. Yannakakis, "Approximate max-flow min-(multi) cut theorems and their applications," 25th STOC, pp. 698-707, 1993.

....partially done at University of S ao Paulo (Brazil) and supported in part by FAPESP (Proc. 96 12111 4) and ProNEx (MCT FINEP) Proj. 107 97) 1 2 Please write authorrunninghead Author Name(s) in file 1. INTRODUCTION Multicommodity Flow problems have been intensely studied for decades [7, 12, 9, 15, 17, 19] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [3, 9, 10, 15, 23] The Weighted Multicut is the ....

....intensely studied for decades [7, 12, 9, 15, 17, 19] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [3, 9, 10, 15, 23]. The Weighted Multicut is the following problem: given an undirected graph G, a weight function w on the edges of G, and a collection of k pairs of distinct vertices (s i ; t i ) of G, find a minimum weight set of edges of G whose removal disconnects each s i from the corresponding t i . The ....

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N. Garg, V. Vazirani and M. Yannakakis. "Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications," SIAM Journal on Computing, 25 (2), 235-251, 1996.

.... Ravi and Rao [9] extend this result to general t q and show that the bound is O(log C log D) where D is the sum of demands (i.e. D = P q2Q t q ) and C is the sum of capacities (i.e. C = P e2E c e ) Tragoudas [20] has improved this bound to O(log jV j log D) and Garg, Vazirani and Yannakakis [5] has further improved it to O(log k log D) where k = jQj. Plotkin and Tardos [18] present the first bound that does not depend on the input data by showing that the upper bound in (3) is at most O(log 2 k) Finally Linial, London and Rabinovich [11] and Aumann and Rabani [1] independently show ....

....from any k 2 K to any one of its sink nodes is at least 1. We next state a O(log k ) bound on the associated min cut max flow ratio. This improves the previous best known bound of O(log k) where k denotes the number of origin destination pairs) presented in Garg, Vazirani and Yannakakis [5]. Lemma 9 Given a maximum multicommodity flow problem, let F denote the maximum total flow, C ( Delta ) denote the capacity of the minimum multicut and k denote the cardinality of the minimal vertex cover of the associated demand graph. If k 1, then c dlog k e C ( Delta ) ....

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N. Garg, V.V. Vazirani, and M. Yannakakis (1993): "Approximate max-flow min-(multi)cut theorems and their applications" in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pp. 698--707.

....we drop any of the three conditions (unweighted, bounded degree, bounded tree width) Finally we show that some of these results extend to the vertex version of Multicut and to a directed version of Multicut. 1 Introduction Multicommodity Flow problems have been intensely studied for decades [FF58, H63, GVY96, KARR90, LR88, PT93] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [DJPSY94, GVY96, GVY97, KARR90, TV93] The Weighted ....

....for decades [FF58, H63, GVY96, KARR90, LR88, PT93] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [DJPSY94, GVY96, GVY97, KARR90, TV93]. The Weighted Multicut is the following problem: given an undirected graph G, a weight function w on the edges of G, and a collection of k pairs of distinct vertices (s i , t i ) of G, find a minimum weight set of edges of G whose removal disconnects each s i from the corresponding t i . The ....

[Article contains additional citation context not shown here]

N. Garg, V. Vazirani and M. Yannakakis. "Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications," SIAM Journal on Computing, 25 (2), 235-251, 1996.

....such approximations [16, 18, 94, 95, 132] and are good candidates for proving strong non approximability. Reasonable approximation factors are typically slow growing functions of the input size like the logarithm. Many known approximation algorithms have poly logarithmic performance guarantees [25, 54, 61, 69, 79, 89, 91, 103, 107, 126, 128, 129, 154]. Good approximation algorithms have constant performance ratios [9, 22, 74, 156, 164] while best possible approximation algorithms achieve guarantees that cannot be better unless P = NP [50, 152] Nearly best possible approximation algorithms have performance guarantees that are within a constant ....

N. Garg, V. Vazirani, and M.Yannakakis, "Approximate max-flow min-(multi) cut theorems and their applications," Proc., 25th Annual ACM Symp. on Theory of Computing, (1993), pp. 698-707.

....paths. Notice that this is in contrast to the general subset feedback set problem that allows an interesting cycle to contain more than one special vertex. Approximating the optimal multicut is a well studied problem, and the best approximation factor that is known in polynomial time is O(log k) [GVY93]. 1.1 Our Results Having established that the subset feedback set problem is NP complete, even for the case of a single special vertex, we consider approximation algorithms. In Section 2, we present our main result: for any weight function on the edges, the subset fes problem can be ....

....denotes the maximum degree in the graph and denotes the value of the optimal fractional subset fvs. The factor of 2 Delta is obtained by reducing the subset fvs problem to the subset fes problem. The factor of 8 ln(jSj 1) is obtained by an application of the graph partitioning technique of [GVY93]. Spheres that are grown around the special vertices, have two properties: i) They lack interesting cycles. ii) The weight of the neighborhood of a sphere can be related to the weight of the sphere. The approximate subset fvs is obtained by taking the union of all the neighborhood sets of the ....

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N. Garg, V.V. Vazirani and M. Yannakakis, "Approximate max-flow min-(multi) cut theorems and their applications," 25th STOC, pp. 698--707, 1993. To appear in SIAM Journal on Computing.

.... hard and simple graph problems: in fact, some variants of MMCF seem to be intimately related with approximation algorithms for several hard graph problems of both theoretical and practical interest [BL84] KAR90] while e approximation combinatorial approaches have been developed for the problem [GVY93] [PT93] Ta94] making MMCF one of few LPs for which an approximation algorithm of practical interest is known [LPS93] In this work, we present some computational experience with a cost decomposition approach to largescale MMCF problems: the idea of applying decomposition methods to MMCF is not ....

N. GARG, V.V. VAZIRANI AND M. YANNAKAKIS "Approximate Max-Flow Min-(Multi)Cut Theorems and their Applications" Proc. 25th STOC, p. 698-707, 1993

....Unfortunately, the latter result does not (and cannot) imply any bounded approximation ratio for the k clustering problem. The hardness of k clustering (for k 3) was later strengthened by Kann et al. [3] The case k = 2 is also known as the minimum edge deletion bipartition problem. Garg et al. [5] gave a O(logN ) approximation algorithm for this case. Supported by Stanford Graduate Fellowship and NSF Grant IIS 9811904 A standard way to reduce the complexity of clustering problems is to assume that the weight function d is a metric. For this case Guttman Beck and Hassin [4] showed a ....

N. Garg, V.V. Vazirani, M. Yannakakis, "Approximate max-flow min-(multi)-cut theorems and their applications", SICOMP (25), 1995, pp. 235-251.

....P = NP , no polynomial time approximation scheme exists if we drop any of the the three conditions: unweighted, bounded degree, bounded tree width. Some of these results extend to the vertex version of Multicut. 1 Introduction Multicommodity Flow problems have been intensely studied for decades [7, 11, 9, 13, 15, 17] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [3, 9, 10, 13, 20] The Weighted Multicut is the ....

....intensely studied for decades [7, 11, 9, 13, 15, 17] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [3, 9, 10, 13, 20]. The Weighted Multicut is the following problem: given an undirected graph G, a weight function w on the edges of G, and a collection of k pairs of distinct vertices (s i ; t i ) of G, find a minimum weight set of edges of G whose removal disconnects each s i from the corresponding t i . ....

[Article contains additional citation context not shown here]

N. Garg, V. Vazirani and M. Yannakakis. "Approximate Max-Flow Min- (Multi)Cut Theorems and Their Applications," SIAM Journal on Computing, 25 (2), 235-251, 1996.

....P = NP . The complexity of a more general G partitioning problem is analyzed in [3] The minimum edge deletion bipartition problem is to find a minimum cost subset E 0 ae E such that (V; EnE 0 ) is bipartite. This is another version of min sum 2 clustering. Garg, Vazirani and Yannakakis [1] gave an O(log jV j) approximation algorithm for this problem, without assuming the triangle inequality. 1 Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv 69978, Israel. fnili,hassing math.tau.ac.il We consider the problem under the assumption that the ....

N. Garg, V.V.Vazirani and M. Yannakakis "Approximate max-flow min-(multi)cut theorems and their applications", SIAM J. Comput. 25, 235-251, 1995.

....binary trees and that it is Max SNP hard if we relax any of the three condition (unweighted, bounded degree, bounded tree width) We also show that some of these results extend to the vertex version of Multicut. 1 Introduction Multicommodity Flow problems have been intensely studied for decades [FF58, H63, GVY96, KARR90, LR88, PT93] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [DJPSY94, GVY96, GVY97, KARR90, TV93] The Weighted ....

....for decades [FF58, H63, GVY96, KARR90, LR88, PT93] because of their practical applications and also of the appealing hardness of several of their versions. The fractional version of a Multicut problem is the dual of a Multicommodity Flow problem and, therefore, Multicut is of similar interest [DJPSY94, GVY96, GVY97, KARR90, TV93]. The Weighted Multicut is the following problem: given an undirected graph G, a weight function w on the edges of G, and a collection of k pairs of distinct vertices (s i ; t i ) of G, find a minimum weight set of edges of G whose removal disconnects each s i from the corresponding t i . The ....

[Article contains additional citation context not shown here]

N. Garg, V. Vazirani and M. Yannakakis. "Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications," SIAM Journal on Computing, 25 (2), 235-251, 1996.

....a decomposition algorithm that does not rely on balanced cuts. His paper proves the existence of feedback vertex sets in unweighted directed graphs of cardinality O( log log log ) Klein et al. KPRT93] considered symmetric multicuts in directed graphs. By extending the work of Garg et al. [GVY93] to the directed case, they obtained an O(log 2 k) approximation factor for this problem. Even et al. ENSS95] considered feedback set problems and their generalizations in directed graphs, as well as multicuts in circular networks. They presented an O(minflog log log ; log n log log n; log ....

.... ; log n log log n; log 2 kg) approximation factor for the subset feedback set problem, where k denotes the number of special vertices. The first two terms in the approximation factor were obtained by extending the work of Seymour [Se95] and the last term was obtained by extending the work of [GVY93]. We use our paradigm to obtain an O(minflog k log log k; log log log g) approximation factor for all of these problems an improvement over previous works for small values of k. The ae separator problem in directed graphs is to find a minimum capacity subset of edges whose removal ....

N. Garg, V.V. Vazirani and M. Yannakakis, "Approximate max-flow min-(multi) cut theorems and their applications," 25th STOC, pp. 698-707, 1993.

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N. Garg, V. Vazirani, M. Yannakakis, "Approximate max-flow min-(multi)cut theorems and their applications," Proc. 25th ACM Symp. on Theory of Computing, 1993, pp. 698--707. 182

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N. Garg, V. Vazirani and M. Yannakakis. "Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications," SIAM Journal on Computing, 25 (2), 235-251, 1996.

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