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We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a near-linear-time 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 [~ over O](m[superscript 3/2])-time maximum flow algorithm of Goldberg and Rao to find an s-t minimum cut in [~ over O](m[superscript 3/2]) time. This corresponds to a (1 + ε)-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 [~ over O](m√n) time. Our algorithm can also be used to improve the running time of sparsest cut approximation algorithms from [~ over O](mn) to [~ over O](n[superscript 2]) and to accelerate several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of random graphs' cut values near their expectations. Our work draws only on elementary probability and graph theory.National Science Foundation (U.S.

Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs