Distributed Computing

, Volume 27, Issue 6, pp 435–443 | Cite as

No sublogarithmic-time approximation scheme for bipartite vertex cover

Article

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 \(\epsilon > 0\) there exists a constant-time distributed algorithm that finds a \((1+\epsilon )\)-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 \(\delta > 0\) 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 (Combinatorica 13:441–454, 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.

Keywords

Distributed graph algorithms Local approximation Lower bounds Vertex cover 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.Helsinki Institute for Information Technology HIITDepartment of Computer ScienceUniversity of HelsinkiFinland

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