DISC 2012 · 26th International Symposium on Distributed Computing, Salvador, Brazil, October 2012 · doi:10.1007/978-3-642-33651-5_13

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 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 *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 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.

Marcos K. Aguilera (Ed.): *Distributed Computing, 26th International Symposium, DISC 2012, Salvador, Brazil, October 16–18, 2012, Proceedings*, volume 7611 of *Lecture Notes in Computer Science*, pages 181–194, Springer, Berlin, 2012

ISBN 978-3-642-33650-8

- Mika Göös and Jukka Suomela: No sublogarithmic-time approximation scheme for bipartite vertex cover ·
*Distributed Computing*27:435–443, 2014