Deterministic Small Vertex Connectivity in Almost Linear Time
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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(√(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.
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