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No Sublogarithmic-Time Approximation Scheme for Bipartite Vertex Cover

  • Mika Göös
  • Jukka Suomela
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7611)

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 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(logn) 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mika Göös
    • 1
  • Jukka Suomela
    • 1
  1. 1.Helsinki Institute for Information Technology HIIT, Department of Computer ScienceUniversity of HelsinkiFinland

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