#### Theory Seminar

# Sorrachai Yingchareonthawornchai: 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, we need to compute the minimum number of vertices that can disconnect the graph 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 talk, we present 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. An interesting aspect that allows our overall algorithm to be efficient is to “lift” several graph problems to hypergraphs and work directly on hypergraphs.

This is joint work with Thatchaphol Saranurak.