  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.
