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

##### Author(s)

Christiano, Paul F.; Kelner, Jonathan Adam; Madry, Aleksander; Spielman, Daniel A.; Teng, Shang-Hua
DownloadKelner_Electrical Flows (arxiv).pdf (757.2Kb)

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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).

##### Date issued

2011-06##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Department of Mathematics##### Journal

Proceedings of the 43rd annual ACM symposium on Theory of Computing, STOC '11

##### Publisher

Association for Computing Machinery

##### Citation

Christiano, Paul et al. “Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs.” Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC '11, ACM Press, 2011. 273.

Version: Author's final manuscript

##### ISBN

978-1-4503-0691-1