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

TL;DR

The paper investigates how s-t vertex connectivity and global vertex connectivity relate across models, revealing almost-tight reductions in parallel and distributed settings and a sharp separation in two-party communication. It introduces common-neighborhood clustering to decompose graphs into CNCs, enabling reductions from global VC to s-t VC with nearly polylog overhead in PRAM and CONGEST. It also proves a near-optimal $ ilde{oldsymbol{Θ}}(n^{1.5})$ two-party communication lower bound for global VC, contrasting with $ ilde{O}(n)$ for s-t VC, and shows a dense-graph reduction from s-t to global VC in PRAM/communication and a matching upper bound. The work also explores streaming implications, providing reductions to maximum bipartite matching that yield subquadratic space and polylog passes, and presents MM-time parallel implementations for VC via matrix-embedding techniques. Overall, the results bridge VC variants across multiple computational paradigms, offering a unified framework (CNC) and delineating both achievable reductions and fundamental limits.

Abstract

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: 1.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 $\tilde O(n^{ω+1})$-work $O(\log^2n)$-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 (when the diameter is moderately small) vertex connectivity algorithm. This answers an open question in [JM STOC'23]. 2. In contrast, we show that global vertex connectivity is strictly harder than s-t vertex connectivity in the two-party communication setting, requiring $\tilde Θ(n^{1.5})$ bits of communication. The s-t variant was known to be solvable in $\tilde O(n)$ 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 \emph{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.

Figures (2)

• Figure 1: All the dashed edges (edges from $V'-\{s\}$ to neighbors of $s$) are deleted. For every $L'$ with $s\in L'$, $N(L')$ does not change.
• Figure 2: Each ellipsoid (and the set $U$) represents a clique. Every node in $V_i$ for $i=1$ to $\sqrt{n}$ has edges to $U$ as illustrated in the graph.

Theorems & Definitions (162)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 1.4
• theorem 1.5
• theorem 1.6
• remark 1.7
• theorem 1.8: Streaming lower bound
• theorem 1.9: Streaming upper bound
• corollary 1.10