Fast Augmenting Paths by Random Sampling from Residual Graphs

Abstract

Consider an $n$-vertex, $m$-edge, undirected graph with integral capacities and max-flow value $v$. We give a new $\tilde{O}(m + nv)$-time maximum flow algorithm. After assigning certain special sampling probabilities to edges in $\tilde{O}(m)$ time, our algorithm is very simple: repeatedly find an augmenting path in a random sample of edges from the residual graph. Breaking from past work, we demonstrate that we can benefit by random sampling from directed (residual) graphs. We also slightly improve an algorithm for approximating flows of arbitrary value, finding a flow of value $(1-\epsilon)$ times the maximum in $\tilde{O}(m\sqrt{n/\epsilon})$ time.