Vertex Connectivity in Poly-logarithmic Max-flows

TL;DR

This paper introduces a reduction from vertex connectivity to maxflow, enabling faster algorithms that significantly improve the previous running time bounds for undirected and directed vertex connectivity problems.

Contribution

The authors present a novel reduction from vertex connectivity to maxflow, leading to the first sub-quadratic time algorithms for vertex connectivity in over 20 years.

Findings

Achieved a vertex connectivity algorithm running in  O(m^{4/3+o(1)}) time.

Improved directed vertex connectivity bound to mn^{1-1/12+o(1)} time.

Established a reduction that leverages maxflow algorithms to solve vertex connectivity efficiently.

Abstract

The vertex connectivity of an $m$-edge $n$-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in $\tilde O(m^{\alpha})$ time for any $\alpha \ge 1$, if there is a $m^{\alpha}$-time maxflow algorithm. Using the current best maxflow algorithm that runs in $m^{4/3+o(1)}$ time (Kathuria, Liu and Sidford, FOCS 2020), this yields a $m^{4/3+o(1)}$-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an $\tilde O(mn)$-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an $o(mn)$ running time was known before our work, even if we assume an $\tilde O(m)$-time maxflow algorithm. Our new technique is robust enough to also improve the best $\tilde O(mn)$-time bound for directed vertex connectivity to $mn^{1-1/12+o(1)}$ time