Results 1  10
of
131
Multicommodity maxflow mincut theorems and their use in designing approximation algorithms
 J. ACM
, 1999
"... Abstract. In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound imp ..."
Abstract

Cited by 374 (6 self)
 Add to MetaCart
Abstract. In this paper, we establish maxflow mincut theorems for several important classes of multicommodity flow problems. In particular, we show that for any nnode multicommodity flow problem with uniform demands, the maxflow for the problem is within an O(log n) factor of the upper bound implied by the mincut. The result (which is existentially optimal) establishes an important analogue of the famous 1commodity maxflow mincut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomialtime (polylog ntimesoptimal) approximation algorithms for wellknown NPhard optimization problems such as graph partitioning, mincut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper.
A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics
 In Proceedings of the 35th Annual ACM Symposium on Theory of Computing
, 2003
"... In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; t ..."
Abstract

Cited by 322 (8 self)
 Add to MetaCart
(Show Context)
In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion#sto n)distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buyatbulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.
Approximate MaxFlow Min(multi)cut Theorems and Their Applications
 SIAM Journal on Computing
, 1993
"... Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us ..."
Abstract

Cited by 163 (3 self)
 Add to MetaCart
(Show Context)
Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed. We prove the following approximate maxflow minmulticut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities. Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut). In addition, the proof technique provides a unified framework in which one can also analyse the case of flows with specified demands, of LeightonRao and Klein et.al., and thereby obtain an improved bound for the latter problem. 1 Introduction Much of flow theory, and the theory of cuts in graphs, is built around a single theorem  the celebrated maxflow mincut theorem of Ford and Fulkerson [FF], and Elias, Feinstein and Shannon [EFS]. The power of this theorem lies in that it relates two fundamental graphtheoretic entities via the potent mechanism of a minmax relation. The importance of this theor...
An O(log k) approximate mincut maxflow theorem and approximation algorithm
 SIAM J. Comput
, 1998
"... Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. A ..."
Abstract

Cited by 140 (6 self)
 Add to MetaCart
Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum concurrent flow for kcommodity flow instances with arbitrary capacities and demands. This improves upon the previously bestknown bound of O(log 2 k) and is existentially tight, up to a constant factor. An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent flow, and thus of the optimal mincut ratio, is presented.
Minimizing Congestion in General Networks
 IN PROCEEDINGS OF THE 43RD IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
, 2002
"... A principle task in parallel and distributed systems is to reduce the communication load in the interconnection network, as this is usually the major bottleneck for the performance of distributed applications. In this paper we introduce a framework for solving online problems that aim to minimize t ..."
Abstract

Cited by 116 (11 self)
 Add to MetaCart
A principle task in parallel and distributed systems is to reduce the communication load in the interconnection network, as this is usually the major bottleneck for the performance of distributed applications. In this paper we introduce a framework for solving online problems that aim to minimize the congestion (i.e. the maximum load of a network link) in general topology networks. We apply this
Improved approximation algorithms for minimum weight vertex separators
 In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS’89
, 1989
"... vertex separators ..."
(Show Context)
Measured descent: A new embedding method for finite metrics
 In Proc. 45th FOCS
, 2004
"... We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for ..."
Abstract

Cited by 99 (32 self)
 Add to MetaCart
(Show Context)
We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any npoint metric space (X, d) embeds in Hilbert space with distortion O ( √ αX · log n), where αX is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O ( √ (log λX)log n) distortion embedding, where λX is the doubling constant of X. Since λX ≤ n, this result recovers Bourgain’s theorem, but when the metric X is, in a sense, “lowdimensional, ” improved bounds are achieved. Our embeddings are volumerespecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volumerespecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted npoint planar graph O(log n) embeds in ℓ∞ with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n) 2). 1
Approximating a Finite Metric by a Small Number of Tree Metrics
 In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science
, 1998
"... Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms f ..."
Abstract

Cited by 95 (10 self)
 Add to MetaCart
(Show Context)
Bartal [4, 5] gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O(log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his
A Polylogarithmic Approximation of the Minimum Bisection
, 2001
"... A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets. ..."
Abstract

Cited by 89 (7 self)
 Add to MetaCart
A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n=2. The bisection cost is the number of edges connecting the two sets.
Approximation algorithms for the 0extension problem
 IN PROCEEDINGS OF THE TWELFTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 2001
"... In the 0extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between t ..."
Abstract

Cited by 70 (3 self)
 Add to MetaCart
(Show Context)
In the 0extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge’s weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis and is closely related to the metric labeling problem introduced by Kleinberg and Tardos. We present approximation algorithms for 0Extension. In arbitrary graphs, we present a O(log k)approximation algorithm, k being the number of terminals. We also give O(1)approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem, previously considered by Karzanov. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis for the multicut problem and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least c √ lg k for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos and they further our understanding on how to use metric relaxations.