Deterministic Vertex Connectivity via Common-Neighborhood Clustering and Pseudorandomness
Yonggang Jiang, Chaitanya Nalam, Thatchaphol Saranurak, Sorrachai Yingchareonthawornchai
TL;DR
The paper delivers a deterministic, near-optimal framework for global minimum vertex cuts in both weighted directed and unweighted undirected graphs. It fuses four core tools—Crossing Families, Common-Neighborhood Clustering, Terminal Reduction, and Selectors—to deterministically compress instances and reduce required max-flow calls, achieving an overall runtime of \widehat{O}(mn) in weighted directed settings (up to subpolynomial factors) and \widehat{O}(mκ) for undirected unweighted graphs. The approach closes longstanding deterministic barriers and answers Gabow's question about O(mn)-time determinism for unweighted graphs, while matching randomized benchmarks up to polylogarithmic factors. The techniques rely on disperser-based crossing families and linear lossless condensers to build compact, pseudorandom structures that preserve cut structure, enabling scalable, exact vertex connectivity computations with provable guarantees. Overall, the work advances deterministic min-cut algorithms and introduces tools with potential applicability to a broader class of graph-structural problems.
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 $\widehat Ω(n^{4})$-time barrier in dense graphs, achievable by trivially computing all-pairs maximum flows. Up to subpolynomial factors, we match the fastest randomized $\tilde O(mn)$-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 $\widehat O(mκ)$-time algorithm where $κ\le n$ is the size of the global minimum vertex cut. For a moderate value of $κ$, this strictly improves upon all previous deterministic algorithms in unweighted graphs with running time $\widehat O(m(n+κ^{2}))$ [Even'75], $\widehat O(m(n+κ\sqrt{n}))$ [Gabow'06], and $\widehat O(m2^{O(κ^{2})})$ [Saranurak and Yingchareonthawornchai'22]. Recently, a linear-time algorithm has been shown by [Korhonen'24] for very small $κ$. 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.