Finding Most Shattering Minimum Vertex Cuts of Polylogarithmic Size in Near-Linear Time

TL;DR

This paper introduces the first near-linear time randomized algorithms for efficiently listing and identifying the most shattering minimum vertex cuts in large graphs, significantly improving over previous quadratic time methods.

Contribution

It presents novel near-linear time algorithms for listing all minimum vertex cuts that create multiple components and for finding the most shattering cut, advancing graph connectivity analysis.

Findings

Breaks the quadratic time barrier for these problems

Uses extended local flow techniques for efficient listing

Employs vertex failure connectivity oracles to speed up algorithms

Abstract

We show the first near-linear time randomized algorithms for listing all minimum vertex cuts of polylogarithmic size that separate the graph into at least three connected components (also known as shredders) and for finding the most shattering one, i.e., the one maximizing the number of connected components. Our algorithms break the quadratic time bound by Cheriyan and Thurimella (STOC'96) for both problems that has stood for more than two decades. Our work also removes a bottleneck to near-linear time algorithms for the vertex connectivity augmentation problem (Jordan '95). Note that it is necessary to list only minimum vertex cuts that separate the graph into at least three components because there can be an exponential number of minimum vertex cuts in general. To obtain near-linear time algorithms, we have extended techniques in local flow algorithms developed by Forster et al. (SODA'20) to list shredders on a local scale. We also exploit fast queries to a pairwise vertex connectivity oracle subject to vertex failures (Long and Saranurak FOCS'22, Kosinas ESA'23). This is the first application of connectivity oracles subject to vertex failures to speed up a static graph algorithm.