Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs

Abstract

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.