Deterministic Small Vertex Connectivity in Almost Linear Time

In the vertex connectivity problem, given an undirected $n$-vertex $m$-edge graph $G$, we need to compute the minimum number of vertices that can disconnect $G$ after removing them. This problem is one of the most well-studied graph problems. From 2019, a new line of work [Nanongkai et al.~STOC'19;SODA'20;STOC'21] has used randomized techniques to break the quadratic-time barrier and, very recently, culminated in an almost-linear time algorithm via the recently announced maxflow algorithm by Chen et al. In contrast, all known deterministic algorithms are much slower. The fastest algorithm [Gabow FOCS'00] takes $O(m(n+\min\{c^{5/2},cn^{3/4}\}))$ time where $c$ is the vertex connectivity. It remains open whether there exists a subquadratic-time deterministic algorithm for any constant $c>3$. In this paper, we give the first deterministic almost-linear time vertex connectivity algorithm for all constants $c$. Our running time is $m^{1+o(1)}2^{O(c^{2})}$ time, which is almost-linear for all $c=o(\sqrt{\log n})$. This is the first deterministic algorithm that breaks the $O(n^{2})$-time bound on sparse graphs where $m=O(n)$, which is known for more than 50 years ago [Kleitman'69]. Towards our result, we give a new reduction framework to vertex expanders which in turn exploits our new almost-linear time construction of mimicking network for vertex connectivity. The previous construction by Kratsch and Wahlstr\"{o}m [FOCS'12] requires large polynomial time and is randomized.