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Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
, 2011
"... We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately t ..."
Abstract

Cited by 17 (0 self)
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We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.
Better Random Sampling Algorithms for Flows in Undirected Graphs
, 1997
"... We present better random sampling algorithms for maximum flows in undirected graphs. Our algorithms apply to capacitated or uncapacitated graphs, and find a maximum flow of value v in ~ O( p mnv) time. This improves on a previous bound of ~ O(m 2=3 n 1=3 v) given by the author recently, which ..."
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Cited by 12 (4 self)
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We present better random sampling algorithms for maximum flows in undirected graphs. Our algorithms apply to capacitated or uncapacitated graphs, and find a maximum flow of value v in ~ O( p mnv) time. This improves on a previous bound of ~ O(m 2=3 n 1=3 v) given by the author recently, which in turn improved on the O(mv) time bound for a typical augmenting path algorithm. In uncapacitated graphs without parallel edges, the bound is no worse than ~ O(n 5=2 ). We give another algorithm that finds a (1 \Gamma ffl) times maximum flow in time ~ O(m p n=ffl), regardless of v. 1 Introduction. Random sampling has been a useful tool for solving cut problems in undirected graphs. In previous work [Kar97a], this author showed that randomly choosing edges from a graph yields a sampled graph in which every cut is close to its expected value with high probability. This led to algorithms for approximating [Kar97a] and exactly finding [Kar96] global mincuts in nearlinear time with hig...