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 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(log n) 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 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.