Directed Global Minimum Cut in Almost-Linear Time

2026-10-07

AI Summary

This study addresses the computational inefficiency of existing algorithms for the global minimum cut in directed graphs, all of which exhibit super-linear time complexity. To overcome this limitation, this work proposes the first near-linear-time algorithm by introducing a randomized reduction from the minimum cut problem to the maximum flow problem. Central to this approach is an iterative directed tree sampling strategy that effectively circumvents the prohibitive overhead associated with traditional approximate packing constructions. By breaking established theoretical lower bounds, the proposed method achieves a time complexity of $m^{1+o(1)}$, significantly improving upon the previous best-known results. This contribution establishes a novel and highly efficient computational paradigm for solving directed graph connectivity problems.

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.

Problem

• directed graph
• global minimum cut
• almost-linear time
• maximum flow

Innovation

• Global Minimum Cut
• Almost-Linear Time
• Iterative Arborescence Sampling
• Maximum Flow
• Directed Graphs