No sublogarithmic-time 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 > 0 there exists a constant-time distributed algorithm that finds a (1 + )-approximation of a maxi-mum matching on 2-coloured graphs of bounded degree. In this work, we show—somewhat surprisingly—that no sublogarithmic-time approxima-tion 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 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, indepen-dent 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.