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 $\tilde O(mn)$-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 $\widehat O(m\kappa)$-time algorithm where $\kappa\le n$ 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], $\widehat O(m(n+\kappa\sqrt{n}))$ [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.

翻译：我们提出了一种确定性算法，用于计算具有 $n$ 个顶点和 $m$ 条边的顶点加权图中全局最小顶点割，其时间复杂度为 $\widehat O(mn)$。该算法打破了稠密图中长期存在的 $\widehat \Omega(n^{4})$ 时间障碍（该障碍可通过平凡计算所有顶点对之间的最大流达到）。在亚多项式因子范围内，我们匹配了 [Henzinger, Rao, and Gabow'00] 提出的最快随机算法 $\tilde O(mn)$，并肯定地回答了 [Gabow'06] 提出的问题：即使在无权图中，是否存在确定性的 $O(mn)$ 时间算法。我们的算法同样适用于有向图。在无权无向图中，我们提出了一种更快的确定性算法，其时间复杂度为 $\widehat O(m\kappa)$，其中 $\kappa\le n$ 为全局最小顶点割的规模。对于中等大小的 $\kappa$，该算法严格优于以往所有无权图中的确定性算法，包括运行时间为 $\widehat O(m(n+\kappa^{2}))$ [Even'75]、$\widehat O(m(n+\kappa\sqrt{n}))$ [Gabow'06] 和 $\widehat O(m2^{O(\kappa^{2})})$ [Saranurak and Yingchareonthawornchai'22] 的算法。最近，[Korhonen'24] 针对极小的 $\kappa$ 提出了一种线性时间算法。我们的方法创新性地应用了由 [Blikstad, Jiang, Mukhopadhyay, Yingchareonthawornchai'25] 近期引入的公共邻域聚类技术，例如在加权图之上和顶点扩展器分解之上进行应用。我们还利用了计算复杂性领域常用的伪随机对象，包括基于 [Wigderson and Zuckerman'99; TaShma, Umans and Zuckerman'01] 中分散器的交叉族，以及基于线性无损压缩器 [Guruswwami, Umans and Vadhan'09; Cheraghchi'11] 的选择器。据我们所知，这是选择器在图算法中的首次应用。