Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
András A. Benczúr, David R. Karger · SIAM Journal on Computing · 2015
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 $\tilde{O}(m^{3/2})$-time maximum flow algorithm of Goldberg and Rao to find an $s$-$t$ minimum cut in $\tilde{O}(n^{3/2})$ time. This corresponds to a $(1+\epsilon)$-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 $\tilde{O}(m\sqrt{n})$ time. Our algorithm can also be used to improve the running time of sparsest cut approximation algorithms from $\tilde{O}(mn)$ to $\tilde{O}(n^2)$ 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.