Almost-Optimal Approximation Algorithms for Global Minimum Cut in Directed Graphs

TL;DR

This paper introduces nearly optimal randomized algorithms for approximating the global minimum cut in directed graphs with improved running times, extending to rooted variants and developing new reductions.

Contribution

It presents the first nearly optimal $(1+psilon)$-approximation algorithms for directed minimum cuts with $O(m^{1+o(1)})$ time, extending techniques and reductions from unweighted to weighted directed graphs.

Findings

Achieved $O(m^{1+o(1)}/psilon)$ running time for approximate minimum cuts.

Extended framework for unweighted to weighted directed graphs.

Developed black-box reduction from general to rooted minimum vertex-cut.

Abstract

We develop new $(1+\epsilon)$-approximation algorithms for finding the global minimum edge-cut in a directed edge-weighted graph, and for finding the global minimum vertex-cut in a directed vertex-weighted graph. Our algorithms are randomized, and have a running time of $O\left(m^{1+o(1)}/\epsilon\right)$ on any $m$-edge $n$-vertex input graph, assuming all edge/vertex weights are polynomially-bounded. In particular, for any constant $\epsilon>0$, our algorithms have an almost-optimal running time of $O\left(m^{1+o(1)}\right)$. The fastest previously-known running time for this setting, due to (Cen et al., FOCS 2021), is $\tilde{O}\left(\min\left\{n^2/\epsilon^2,m^{1+o(1)}\sqrt{n}\right\}\right)$ for Minimum Edge-Cut, and $\tilde{O}\left(n^2/\epsilon^2\right)$ for Minimum Vertex-Cut. Our results further extend to the rooted variants of the Minimum Edge-Cut and Minimum Vertex-Cut problems, where the algorithm is additionally given a root vertex $r$, and the goal is to find a minimum-weight cut separating any vertex from the root $r$. In terms of techniques, we build upon and extend a framework that was recently introduced by (Chuzhoy et al., SODA 2026) for solving the Minimum Vertex-Cut problem in unweighted directed graphs. Additionally, in order to obtain our result for the Global Minimum Vertex-Cut problem, we develop a novel black-box reduction from this problem to its rooted variant. Prior to our work, such reductions were only known for more restricted settings, such as when all vertex-weights are unit.