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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.

Submission history

From: Ron Mosenzon [view email]
[v1] Tue, 9 Dec 2025 19:51:45 UTC (67 KB)
[v2] Tue, 16 Dec 2025 02:38:55 UTC (67 KB)
[v3] Mon, 20 Jul 2026 20:39:03 UTC (67 KB)