Vertex Connectivity in Poly-logarithmic Max-flows

Understand

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.