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We introduce a new approach to computing an approximately maximum s-t flow ina capacitated, undirected graph. This flow is computed by solving a sequence ofelectrical flow problems. Each electrical flow is given by the solution of asystem of linear equations in a Laplacian matrix, and thus may be approximatelycomputed in nearly-linear time.  Using this approach, we develop the fastest known algorithm for computingapproximately maximum s-t flows. For a graph having n vertices and m edges, ouralgorithm 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 thisproblem as well. Previously, the best dependence on m and n was achieved by thealgorithm of Goldberg and Rao (J. ACM 1998), which can be used to computeapproximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), andapproximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3}).

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