Jukka Suomela: Publications: No sublogarithmic-time approximation scheme for bipartite vertex cover
|Authors:||Mika Göös and Jukka Suomela|
|Title:||No sublogarithmic-time approximation scheme for bipartite vertex cover|
|Conference:||26th International Symposium on Distributed Computing (DISC), Salvador, Brazil, October 2012|
|Proceedings:||Marcos K. Aguilera (Ed.): Distributed Computing. 26th International Symposium, DISC 2012. Salvador, Brazil, October 16–18, 2012. Proceedings|
|Lecture Notes in Computer Science, volume 7611
Springer, Berlin, Germany, 2012
|Links:||publisher’s version · authors’ version
© Springer 2012 — The original publication is available at www.springerlink.com.
|presentation, 17 October 2012|
|conference · proceedings · series · DBLP · arXiv.org|
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.