Deterministic Small Vertex Connectivity in Almost Linear Time - Saranurak, Thatchaphol; Yingchareonthawornchai, Sorrachai (Proceedings / annual Symposium on Foundations of Computer Science, 2022)
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 Wahlström [FOCS'12] requires large polynomial time and is randomized.