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 (m^α) time for any α ≥ 1, if there is a m^α-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 Õ(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 (m)-time maxflow algorithm.

Our new technique is robust enough to also improve the best Õ(mn)-time bound for directed vertex connectivity to mn^{1−1/12+o(1)} time