No Sublogarithmic-Time Approximation Scheme for Bipartite Vertex Cover
Jukka Suomela · 2013
Abstract König’s theorem states that on bipartite graphs the size of a maximum matching equals the size of a mini-mum vertex cover. It is known from prior work that for every ε> 0 there exists a constant-time distributed algorithm that finds a (1+ ε)-approximation of a maximum matching on bounded-degree graphs. In this work, we show—somewhat surprisingly—that no sublogarithmic-time approximation scheme exists for the dual problem: there is a constant δ> 0 so that no randomised distributed algorithm with running time o(logn) can find a (1+ δ)-approximation of a mini-mum vertex cover on 2-coloured graphs of maximum de-gree 3. In fact, a simple application of the Linial–Saks (1993) decomposition demonstrates that this run-time 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.