No Sublogarithmictime Approximation Scheme for Bipartite Vertex Cover
Abstract
König'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 constanttime distributed algorithm that finds a (1+\epsilon)approximation of a maximum matching on 2coloured graphs of bounded degree. In this work, we showsomewhat surprisinglythat no sublogarithmictime 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 2coloured graphs of maximum degree 3. In fact, a simple application of the LinialSaks (1993) decomposition demonstrates that this lower bound is tight. Our lowerbound 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2012
 arXiv:
 arXiv:1205.4605
 Bibcode:
 2012arXiv1205.4605G
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms
 EPrint:
 11 pages, 5 figures