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

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- )approximately maximum s-t flow in time1 O mn1/3 -11/3 . A dual version of our approach computes a (1 + )-approximately minimum s-t cut in time O m + n4/3 -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 O m n -1 , and approximately minimum s-t cuts in time O m + n3/2 -3 . ...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, Shang-Hua Teng
• Keywords: CORR 2010, Education, Electrical Flow, Fastest Known Algorithm, Maximum S-t Flows