Electrical flows, Laplacian systems, and faster approximation of maximum flow in undirected graphs

Abstract

We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearly-linear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum s-t flows. For a graph having n vertices and m edges, our algorithm computes a (1-ε)-approximately maximum s-t flow in time ~O(mn1/3ε-11/3). A dual version of our approach gives the fastest known algorithm for computing a (1+ε)-approximately minimum s-t cut. It takes ~O(m+n4/3ε-16/3) time. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum s-t flows in time ~O({m√nε-1), and approximately minimum s-t cuts in time ~O(m+n3/2ε-3).