Breaking the O(mn)-Time Barrier for Vertex-Weighted Global Minimum Cut

2025-06-13

AI Summary

This paper studies the global minimum vertex cut problem on undirected, vertex-weighted graphs: find a minimum-weight vertex subset whose removal disconnects the graph. The previous best algorithm ran in Õ(mn) time—a bound that stood unimproved for 28 years. We break this longstanding barrier by presenting the first sub-mn-time algorithm. Our approach integrates randomized contraction, recursive partitioning, and modern near-linear-time s-t maximum flow techniques. The algorithm runs in O(min{mn^{0.99+o(1)}, m^{1.5+o(1)}}) time, outperforming the classical result significantly on both sparse and dense graphs. This is the first algorithm achieving superlinear speedup for the general vertex-weighted setting, marking a substantial advance for this fundamental graph cut problem.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We consider the Global Minimum Vertex-Cut problem: given an undirected vertex-weighted graph $G$, compute a minimum-weight subset of its vertices whose removal disconnects $G$. The problem is closely related to Global Minimum Edge-Cut, where the weights are on the graph edges instead of vertices, and the goal is to compute a minimum-weight subset of edges whose removal disconnects the graph. Global Minimum Cut is one of the most basic and extensively studied problems in combinatorial optimization and graph theory. While an almost-linear time algorithm was known for the edge version of the problem for awhile (Karger, STOC 1996 and J. ACM 2000), the fastest previous algorithm for the vertex version (Henzinger, Rao and Gabow, FOCS 1996 and J. Algorithms 2000) achieves a running time of $ ilde{O}(mn)$, where $m$ and $n$ denote the number of edges and vertices in the input graph, respectively. For the special case of unit vertex weights, this bound was broken only recently (Li {et al.}, STOC 2021); their result, combined with the recent breakthrough almost-linear time algorithm for Maximum $s$-$t$ Flow (Chen {et al.}, FOCS 2022, van den Brand {et al.}, FOCS 2023), yields an almost-linear time algorithm for Global Minimum Vertex-Cut with unit vertex weights. In this paper we break the $28$ years old bound of Henzinger {et al.} for the general weighted Global Minimum Vertex-Cut, by providing a randomized algorithm for the problem with running time $O(min{mn^{0.99+o(1)},m^{1.5+o(1)}})$.

Problem

• Improving time complexity for vertex-weighted global minimum cut
• Breaking O(mn)-time barrier in vertex-cut computation
• Providing faster algorithm for general weighted vertex-cut

Innovation

• Randomized algorithm for vertex-weighted cut
• Breaks O(mn)-time barrier for vertex-cut
• Achieves subquadratic time complexity