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

TL;DR

This work advances almost-linear-time algorithms for directed global minimum cuts by presenting $(1+\varepsilon)$-approximation randomized methods for both edge- and vertex-weighted versions, with running times near $O(m^{1+o(1)})$ under polynomial weight bounds. Central to the approach is a rooted-minimum-cut framework that uses hierarchical terminal sparsification (BatchSparsify) and access to fast $s$-$t$ flow/cut subroutines, yielding efficient approximations for rooted problems. A key contribution is a novel black-box reduction that transforms Global Minimum Vertex-Cut into Rooted Minimum Vertex-Cut instances with total sparsified size $O(m)$, enabling a combined solution for the global vertex-cut via rooted subproblems. The results subsume prior work and align with concurrent independent efforts, while providing a robust toolkit (sparsification, sampling of terminals, and weight-dependence reduction) that may inform future exact and approximate algorithms in directed graphs. Overall, the paper delivers near-optimal time guarantees for fundamental cuts in directed graphs and introduces versatile reductions that broaden the applicability of rooted-cut techniques to global problems.

Abstract

We develop new $(1+ε)$-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 $O\left(m^{1+o(1)}/ε\right)$ on any $m$-edge $n$-vertex input graph, assuming all edge/vertex weights are polynomially-bounded. In particular, for any constant $ε>0$, 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 $\tilde{O}\left(\min\left\{n^2/ε^2,m^{1+o(1)}\sqrt{n}\right\}\right)$ for Minimum Edge-Cut, and $\tilde{O}\left(n^2/ε^2\right)$ 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.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3: Rooted Minimum Vertex-Cut Oracle
• Theorem 1.4: Reduction from Global to Rooted Minimum Vertex-Cut
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: vdBCK23
• Lemma 2.4: vdBCK23