Global vs. s-t Vertex Connectivity Beyond Sequential: Almost-Perfect Reductions and Near-Optimal Separations

A recent breakthrough by [LNPSY STOC'21] showed that solving s-t vertex connectivity is sufficient (up to polylogarithmic factors) to solve (global) vertex connectivity in the sequential model. This raises a natural question: What is the relationship between s-t and global vertex connectivity in other computational models? In this paper, we demonstrate that the connection between global and s-t variants behaves very differently across computational models. In parallel and distributed models, we obtain almost tight reductions from global to s-t vertex connectivity. In PRAM, this leads to a $n^{ω+o(1)}$-work and $n^{o(1)}$-depth algorithm for vertex connectivity, improving over the 35-year-old $Õ(n^{ω+1})$-work $O(\mathrm{log}^2n)$-depth algorithm by [LLW FOCS'86], where $ω$ is the matrix multiplication exponent and $n$ is the number of vertices. In CONGEST, the reduction implies the first sublinear-round vertex connectivity algorithm when the diameter is moderately small. This answers an open question in [JM STOC'23]. In contrast, we show that global vertex connectivity is strictly harder than s-t vertex connectivity in the two-party communication setting, requiring $n^{1.5}$ bits of communication. The s-t variant was known to be solvable in $Õ(n)$ communication [BvdBEMN FOCS'22]. Our results resolve open problems raised by [MN STOC'20, BvdBEMN FOCS'22, AS SOSA'23]. At the heart of our results is a new graph decomposition framework we call common-neighborhood clustering, which can be applied in multiple models. Finally, we observe that global vertex connectivity cannot be solved without using s-t vertex connectivity by proving an s-t to global reduction in dense graphs in the PRAM and communication models.