Vertex Connectivity in Poly-logarithmic Max-flows

03/31/2021
by   Jason Li, et al.
0

The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in Õ(m^α) time for any α≥ 1, if there is a m^α-time maxflow algorithm. Using the current best maxflow algorithm that runs in m^4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m^4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an Õ(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an Õ(m)-time maxflow algorithm. Our new technique is robust enough to also improve the best Õ(mn)-time bound for directed vertex connectivity to mn^1-1/12+o(1) time

READ FULL TEXT

page 1

page 2

page 3

page 4

research
11/03/2021

Augmenting Edge Connectivity via Isolating Cuts

We give an algorithm for augmenting the edge connectivity of an undirect...
research
10/23/2019

Vertex Sparsifiers for c-Edge Connectivity

We show the existence of O(f(c)k) sized vertex sparsifiers that preserve...
research
08/09/2023

Deterministic k-Vertex Connectivity in k^2 Max-flows

An n-vertex m-edge graph is k-vertex connected if it cannot be disconnec...
research
07/14/2020

Component Order Connectivity in Directed Graphs

A directed graph D is semicomplete if for every pair x,y of vertices of ...
research
05/17/2021

A New Vertex Connectivity Metric

A new metric for quantifying pairwise vertex connectivity in graphs is d...
research
06/25/2020

Augmenting the Algebraic Connectivity of Graphs

For any undirected graph G=(V,E) and a set E_W of candidate edges with E...
research
08/19/2020

A Simple Deterministic Algorithm for Edge Connectivity

We show a deterministic algorithm for computing edge connectivity of a s...

Please sign up or login with your details

Forgot password? Click here to reset