Minimum Cuts in Directed Graphs via $\sqrt{n}$ Max-Flows

TL;DR

This paper presents a new algorithm for finding minimum cuts in directed graphs that significantly reduces the number of max-flow computations needed, leading to faster overall runtimes.

Contribution

The authors introduce an algorithm that finds mincuts in directed graphs with only () max-flow calls, improving the classic (mn) bound for this problem.

Findings

Achieves () max-flow calls for mincut detection.

Yields an (m + n^2) time algorithm with state-of-the-art maxflow methods.

Improves the 30-year-old runtime bound for directed mincut problems.

Abstract

We give an algorithm to find a mincut in an $n$-vertex, $m$-edge weighted directed graph using $\tilde O(\sqrt{n})$ calls to any maxflow subroutine. Using state of the art maxflow algorithms, this yields a directed mincut algorithm that runs in $\tilde O(m\sqrt{n} + n^2)$ time. This improves on the 30 year old bound of $\tilde O(mn)$ obtained by Hao and Orlin for this problem.