No Sublogarithmic-time Approximation Scheme for Bipartite Vertex Cover

TL;DR

This paper proves that, unlike maximum matching, there is no sublogarithmic-time distributed approximation scheme for minimum vertex cover in bipartite graphs, highlighting a fundamental complexity difference.

Contribution

It establishes a tight lower bound showing the impossibility of sublogarithmic-time approximation for bipartite vertex cover, contrasting prior results for maximum matching.

Findings

No sublogarithmic-time distributed approximation for bipartite vertex cover.

The lower bound is tight, demonstrated via Linial--Saks decomposition.

A related cut minimisation problem is also shown to be hard to approximate locally.

Abstract

K\"onig's theorem states that on bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. It is known from prior work that for every \epsilon > 0 there exists a constant-time distributed algorithm that finds a (1+\epsilon)-approximation of a maximum matching on 2-coloured graphs of bounded degree. In this work, we show---somewhat surprisingly---that no sublogarithmic-time approximation scheme exists for the dual problem: there is a constant \delta > 0 so that no randomised distributed algorithm with running time o(\log n) can find a (1+\delta)-approximation of a minimum vertex cover on 2-coloured graphs of maximum degree 3. In fact, a simple application of the Linial--Saks (1993) decomposition demonstrates that this lower bound is tight. Our lower-bound construction is simple and, to some extent, independent of previous techniques. Along the way we prove that a certain cut minimisation problem, which might be of independent interest, is hard to approximate locally on expander graphs.