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arXiv · 2510.17182

Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs

Aaron Bernstein·Joakim Blikstad·Jason Li·Thatchaphol Saranurak·Ta-Wei Tu
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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.

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Aaron Bernstein, Joakim Blikstad, Jason Li, Thatchaphol Saranurak, Ta-Wei Tu. 2025-10-20. Combinatorial Maximum Flow via Weighted Push-Relabel on Shortcut Graphs. https://arxiv.org/abs/2510.17182

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