[Theory] UC Theory Seminar

[Alexander Razborov]

Departments of Mathematics & Computer Science
Combinatorics & Theory Seminar

Tuesday, October 11, 3:30pm
Location Кеnt 107

SPEAKER: Sorrachai Yingchareonthawornchai (Aalto University, Finland)

TITLE: Deterministic Small Vertex Connectivity in Almost Linear Time

ABSTRACT: 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. 

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