A Fast Max Flow Algorithm

TL;DR

This paper introduces a new variant of the excess scaling algorithm for max flow that outperforms previous algorithms in certain graph classes, especially when the number of edges is close to linear in the number of nodes.

Contribution

The paper presents a novel max flow algorithm variant that improves upon existing algorithms' running times for specific graph densities.

Findings

New max flow algorithm dominates previous algorithms in certain graph classes.

For graphs with m=O(n log n), the algorithm is faster by a factor of O(log log n).

The algorithm's running time strictly improves over King et al.'s method.

Abstract

In 2013, Orlin proved that the max flow problem could be solved in $O(nm)$ time. His algorithm ran in $O(nm + m^{1.94})$ time, which was the fastest for graphs with fewer than $n^{1.06}$ arcs. If the graph was not sufficiently sparse, the fastest running time was an algorithm due to King, Rao, and Tarjan. We describe a new variant of the excess scaling algorithm for the max flow problem whose running time strictly dominates the running time of the algorithm by King et al. Moreover, for graphs in which $m = O(n \log n)$, the running time of our algorithm dominates that of King et al. by a factor of $O(\log\log n)$.