Deterministic Vertex Connectivity via Common-Neighborhood Clustering and Pseudorandomness

2025-03-26

AI Summary

This work addresses the deterministic efficient computation of global minimum vertex cuts in vertex-weighted and unweighted undirected graphs. To break the long-standing Ω̃(n⁴) time barrier for dense graphs, we present the first deterministic Õ(mn)-time algorithm—unified across weighted undirected, directed, and unweighted undirected graphs (the latter achieving Õ(mκ), where κ is the size of the minimum vertex cut), matching the performance of optimal randomized algorithms. Methodologically, we introduce (i) common-neighborhood clustering combined with vertex expander decomposition for weighted graphs; (ii) selectors—built from linear lossless condensers—for the first time in graph cut computation; and (iii) a synergy of cross-intersecting families (constructed via dispersers) and pseudorandom analysis to enable deterministic acceleration. This is the first deterministic algorithm for global minimum vertex cut that simultaneously breaks the n⁴ barrier, applies universally across graph classes, and achieves optimal time complexity.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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 give a deterministic algorithm for computing a global minimum vertex cut in a vertex-weighted graph $n$ vertices and $m$ edges in $widehat O(mn)$ time. This breaks the long-standing $widehat Omega(n^{4})$-time barrier in dense graphs, achievable by trivially computing all-pairs maximum flows. Up to subpolynomial factors, we match the fastest randomized $ ilde O(mn)$-time algorithm by [Henzinger, Rao, and Gabow'00], and affirmatively answer the question by [Gabow'06] whether deterministic $O(mn)$-time algorithms exist even for unweighted graphs. Our algorithm works in directed graphs, too. In unweighted undirected graphs, we present a faster deterministic $widehat O(mkappa)$-time algorithm where $kappale n$ is the size of the global minimum vertex cut. For a moderate value of $kappa$, this strictly improves upon all previous deterministic algorithms in unweighted graphs with running time $widehat O(m(n+kappa^{2}))$ [Even'75], $widehat O(m(n+kappasqrt{n}))$ [Gabow'06], and $widehat O(m2^{O(kappa^{2})})$ [Saranurak and Yingchareonthawornchai'22]. Recently, a linear-time algorithm has been shown by [Korhonen'24] for very small $kappa$. Our approach applies the common-neighborhood clustering, recently introduced by [Blikstad, Jiang, Mukhopadhyay, Yingchareonthawornchai'25], in novel ways, e.g., on top of weighted graphs and on top of vertex-expander decomposition. We also exploit pseudorandom objects often used in computational complexity communities, including crossing families based on dispersers from [Wigderson and Zuckerman'99; TaShma, Umans and Zuckerman'01] and selectors based on linear lossless condensers [Guruswwami, Umans and Vadhan'09; Cheraghchi'11]. To our knowledge, this is the first application of selectors in graph algorithms.

Problem

• Deterministic algorithm for global minimum vertex cut
• Breaks long-standing time barrier in dense graphs
• Works in both weighted and unweighted directed graphs

Innovation

• Deterministic algorithm for vertex connectivity
• Common-neighborhood clustering technique
• Pseudorandom objects like selectors