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...
| Main Authors: | , , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Association for Computing Machinery
2012
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| Online Access: | http://hdl.handle.net/1721.1/71698 https://orcid.org/0000-0003-0536-0323 https://orcid.org/0000-0002-4257-4198 |
| Summary: | 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). |
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