A New Push-Relabel Algorithm for the Maximum Flow Problem

In this paper, we present a faster push-relabel algorithm for the maximum flow problem on bounded-degree networks with n vertices and m arcs. We show how to compute a maximum flow in O(mn) time. This matches the results of Orlin’s algorithm, which runs in O(mn + m log n) time on general networks (and O(mn) time on bounded-degree networks). Our main result is improving on the generic push-relabel algorithm (Goldberg & Tarjan, 1988) by reducing the number of nonsaturating pushes to O(mn) across all scaling phases. This improvement is reached by a novel combination of Ahuja and Orlin’s excess scaling method and Orlin’s compact flow networks (STOC ‘13). A major contribution of this paper is demonstrating that the compact networks technique can be extended to the push-relabel family of algorithms. Moreover, we show that an extension of our result to general networks would imply an algorithm that runs in O(mn) time, when m = O(n2−ε). An algorithm incorporating some or all of our techniques may be a promising avenue towards an O(mn)-time algorithm for all edge densities.