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

  • Authors:

  • Paul Christiano;Jonathan A. Kelner;Aleksander Madry;Daniel A. Spielman;Shang-Hua Teng

  • Affiliations:

  • MIT, Cambridge, MA, USA;MIT, Cambridge, MA, USA;MIT, Cambridge, MA, USA;Yale University, New Haven, CT, USA;USC, Los Angeles, CA, USA

  • Venue:
  • Proceedings of the forty-third annual ACM symposium on Theory of computing
  • Year:
  • 2011

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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).