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

2025-03-26

AI Summary

This paper systematically investigates the computational relationship between global vertex connectivity and $s$-$t$ vertex connectivity across parallel (PRAM), distributed (CONGEST), and two-party communication models. Methodologically, it introduces a novel graph decomposition framework—“common-neighbor clustering”—and integrates matrix multiplication acceleration with reduction techniques. The contributions are threefold: (1) It establishes the first near-tight cross-model reductions, proving computational equivalence in PRAM and CONGEST, while showing strict separation in the two-party model via a $widetilde{Theta}(n^{1.5})$ lower bound for global connectivity; (2) It achieves an $n^{omega+o(1)}$-work PRAM algorithm, nearly optimal under current matrix multiplication bounds; (3) It delivers the first sublinear-round algorithm for vertex connectivity in CONGEST. Collectively, these results transcend sequential-model assumptions and provide a unified characterization of the intrinsic complexity of vertex connectivity across all three fundamental computational models.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $n^{omega+o(1)}$-work and $n^{o(1)}$-depth algorithm for vertex connectivity, improving over the 35-year-old $ ilde O(n^{omega+1})$-work $O(log^2n)$-depth algorithm by [LLW FOCS'86], where $omega$ 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 $ ilde Theta (n^{1.5})$ bits of communication. The s-t variant was known to be solvable in $ ilde 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.

Problem

• Explore relationship between s-t and global vertex connectivity across models
• Develop almost tight reductions in parallel and distributed models
• Show global connectivity is harder than s-t in communication settings

Innovation

• Almost tight reductions in parallel and distributed models
• New graph decomposition framework: common-neighborhood clustering
• Proves global connectivity harder in two-party communication