Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs

TL;DR

This paper presents a simple, near-optimal combinatorial algorithm for exact maximum flow in dense directed graphs, improving previous results and providing the first deterministic near-linear time solution for vertex-capacitated max flow.

Contribution

The authors introduce a simplified combinatorial algorithm for maximum flow that improves upon recent results and derandomizes key components for deterministic near-linear time performance.

Findings

Achieves near-linear time complexity $ ilde{O}(n^{2} ext{log } U)$ for dense graphs.

Provides a full C++ implementation demonstrating simplicity.

First deterministic near-linear time algorithm for vertex-capacitated max flow.

Abstract

We give a combinatorial algorithm for computing exact maximum flows in directed graphs with $n$ vertices and edge capacities from $\{1,\dots,U\}$ in $\tilde{O}(n^{2}\log U)$ time, which is near-optimal on dense graphs. This shaves an $n^{o(1)}$ factor from the recent result of [Bernstein-Blikstad-Saranurak-Tu FOCS'24] and, more importantly, greatly simplifies their algorithm. We believe that ours is by a significant margin the simplest of all algorithms that go beyond $\tilde{O}(m\sqrt{n})$ time in general graphs. To highlight this relative simplicity, we provide a full implementation of the algorithm in C++. The only randomized component of our work is the cut-matching game. Via existing tools, we show how to derandomize it for vertex-capacitated max flow and obtain a deterministic $\tilde{O}(n^2)$ time algorithm. This marks the first deterministic near-linear time algorithm for this problem (or even for the special case of bipartite matching) in any density regime.