Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs

TL;DR

This paper presents a novel approach using electrical flows and Laplacian systems to compute approximate maximum s-t flows and minimum s-t cuts in undirected graphs more efficiently than previous methods.

Contribution

It introduces a new algorithm that leverages electrical flow computations via Laplacian systems to achieve faster approximate max flow and min cut algorithms.

Findings

Fastest known approximate max s-t flow algorithm with time ten;O(mn^{1/3}\u03b5^{-11/3})

Fastest known approximate min s-t cut algorithm with time ten;O(m + n^{4/3}\u03b5^{-8/3})

Improves upon previous algorithms with better dependence on m and n.

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-\epsilon)-approximately maximum s-t flow in time \tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a (1+\epsilon)-approximately minimum s-t cut in time \tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this problem as well. 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 \tilde{O}(m\sqrt{n}\epsilon^{-1}), and approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3}).