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No sublogarithmic-time approximation scheme for bipartite vertex cover

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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.

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Acknowledgments

Many thanks to Valentin Polishchuk for discussions, and to anonymous reviewers for their helpful comments and suggestions. This work was supported in part by the Academy of Finland, Grants 132380 and 252018.

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, 3302-10 King’s College Road, Toronto, ON , M5S 3G4, Canada

    Mika Göös

  2. Helsinki Institute for Information Technology HIIT, Department of Computer Science, P.O. Box 68, 00014 , University of Helsinki, Finland

    Jukka Suomela

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  1. Mika Göös
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  2. Jukka Suomela
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Correspondence to Mika Göös.

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A preliminary version of this work [5] appeared in DISC 2012.

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Göös, M., Suomela, J. No sublogarithmic-time approximation scheme for bipartite vertex cover. Distrib. Comput. 27, 435–443 (2014). https://doi.org/10.1007/s00446-013-0194-z

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  • DOI: https://doi.org/10.1007/s00446-013-0194-z

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