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arXiv · 2610.10898

Directed Global Minimum Cut in Almost-Linear Time

Henry Fleischmann·Jason Li·Thatchaphol Saranurak·Benyu Wang
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Abstract

We give a randomized reduction from global minimum cut in a directed graph with $n$ vertices and $m$ weighted edges to maximum-flow computations on graphs of total size $\widetilde{O}(m)$. For polynomially bounded integral weights, this yields the first almost-linear $m^{1+o(1)}$-time algorithm [vdBCK+23], improving the previous $m^{1+o(1)}\min\{\sqrt{n},n/m^{1/3}\}$ bound [CLN+22]. Previous work reduced finding the minimum cut to constructing a 1-respecting arborescence, a directed spanning tree with exactly one edge crossing a minimum cut [CLN+22]. They then construct such a tree by sampling from a $(1+ε)$-approximate arborescence packing, which currently requires superlinear time. Our algorithm sidesteps constructing the packing entirely and relies on a novel iterative arborescence sampling procedure. We show that, after $O(\log n)$ rounds of sampling, our final arborescence 1-respects the minimum cut with constant probability.

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Henry Fleischmann, Jason Li, Thatchaphol Saranurak, Benyu Wang. 2026-10-09. Directed Global Minimum Cut in Almost-Linear Time. https://arxiv.org/abs/2610.10898

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