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Max flows in O(nm) time, or better (2012)

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by James B. Orlin
Citations:7 - 0 self
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BibTeX

@MISC{Orlin12maxflows,
    author = {James B. Orlin},
    title = {Max flows in O(nm) time, or better},
    year = {2012}
}

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Abstract

In this paper, we present improved polynomial time algorithms for the max flow problem defined on a network with n nodes and m arcs. We show how to solve the max flow problem in O(nm) time, improving upon the best previous algorithm due to King, Rao, and Tarjan, who solved the max flow problem in O(nm log m/(n log n) n) time. In the case that m = O(n), we improve the running time to O(n 2 / log n). We further improve the running time in the case that U ∗ = Umax/Umin is not too large, where Umax denotes the largest finite capacity and Umin denotes the smallest non-zero capacity. If log(U ∗ ) = O(n 1/3 log −3 n), we show how to solve the max flow problem in O(nm / log n) steps. In the case that log(U ∗ ) = O(log k n) for some fixed positive integer k, we show how to solve the max flow problem in Õ(n8/3) time. This latter algorithm relies on a subroutine for fast matrix multiplication. 1

Keyphrases

max flow problem    nm log    running time    polynomial time algorithm    latter algorithm relies    finite capacity    fixed positive integer    non-zero capacity    previous algorithm    fast matrix multiplication    umax umin   

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