Max flows in O(nm) time, or better

Abstract

In this paper, we present improved polynomial time algorithms for the max flow problem defined on sparse networks with n nodes and m arcs. We show how to solve the max flow problem in O(nm + m[superscript 31/16] log[superscript 2] n) time. In the case that m = O(n[superscript 1.06]), this improves upon the best previous algorithm due to King, Rao, and Tarjan, who solved the max flow problem in O(nm logm/(n log n)n) time. This establishes that the max flow problem is solvable in O(nm) time for all values of n and m. In the case that m = O(n), we improve the running time to O(n[superscript 2]/ log n).