An $n$-vertex $m$-edge graph is \emph{$k$-vertex connected} if it cannot be disconnected by deleting less than $k$ vertices. After more than half a century of intensive research, the result by [Li et al. STOC'21] finally gave a \emph{randomized} algorithm for checking $k$-connectivity in near-optimal $\widehat{O}(m)$ time. (We use $\widehat{O}(\cdot)$ to hide an $n^{o(1)}$ factor.) Deterministic algorithms, unfortunately, have remained much slower even if we assume a linear-time max-flow algorithm: they either require at least $\Omega(mn)$ time [Even'75; Henzinger Rao and Gabow, FOCS'96; Gabow, FOCS'00] or assume that $k=o(\sqrt{\log n})$ [Saranurak and Yingchareonthawornchai, FOCS'22]. We show a \emph{deterministic} algorithm for checking $k$-vertex connectivity in time proportional to making $\widehat{O}(k^{2})$ max-flow calls, and, hence, in $\widehat{O}(mk^{2})$ time using the deterministic max-flow algorithm by [Brand et al. FOCS'23]. Our algorithm gives the first almost-linear-time bound for all $k$ where $\sqrt{\log n}\le k\le n^{o(1)}$ and subsumes up to a sub polynomial factor the long-standing state-of-the-art algorithm by [Even'75] which requires $O(n+k^{2})$ max-flow calls. Our key technique is a deterministic algorithm for terminal reduction for vertex connectivity: given a terminal set separated by a vertex mincut, output either a vertex mincut or a smaller terminal set that remains separated by a vertex mincut. We also show a deterministic $(1+\epsilon)$-approximation algorithm for vertex connectivity that makes $O(n/\epsilon^2)$ max-flow calls, improving the bound of $O(n^{1.5})$ max-flow calls in the exact algorithm of [Gabow, FOCS'00]. The technique is based on Ramanujan graphs.

翻译：一个具有 $n$ 个顶点和 $m$ 条边的图被称为 \emph{$k$-顶点连通}，如果删除少于 $k$ 个顶点不会使其不连通。经过半个多世纪的深入研究，[Li等, STOC'21]的结果最终给出了一种用于检查 $k$-连通性的 \emph{随机化} 算法，其运行时间为接近最优的 $\widehat{O}(m)$。（我们使用 $\widehat{O}(\cdot)$ 来隐藏 $n^{o(1)}$ 因子。）不幸的是，即使假设存在线性时间的最大流算法，确定性算法仍然慢得多：它们要么需要至少 $\Omega(mn)$ 的时间 [Even'75; Henzinger Rao和Gabow, FOCS'96; Gabow, FOCS'00]，要么假设 $k=o(\sqrt{\log n})$ [Saranurak和Yingchareonthawornchai, FOCS'22]。我们展示了一种用于检查 $k$-顶点连通性的 \emph{确定性} 算法，其时间与进行 $\widehat{O}(k^{2})$ 次最大流调用成正比，因此，使用[Brand等, FOCS'23]的确定性最大流算法，其时间为 $\widehat{O}(mk^{2})$。对于所有满足 $\sqrt{\log n}\le k\le n^{o(1)}$ 的 $k$，我们的算法首次给出了几乎线性的时间界，并在多项式因子上超越了长期以来的最先进算法[Even'75]，该算法需要 $O(n+k^{2})$ 次最大流调用。我们的关键技术是一种用于顶点连通性中终端缩减的确定性算法：给定一个由顶点最小割分隔的终端集，输出一个顶点最小割或一个仍然由顶点最小割分隔的更小终端集。我们还展示了一种用于顶点连通性的确定性 $(1+\epsilon)$-近似算法，该算法进行 $O(n/\epsilon^2)$ 次最大流调用，改进了[Gabow, FOCS'00]精确算法中 $O(n^{1.5})$ 次最大流调用的界。该技术基于拉马努金图。