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Vertex Cover in Graphs with Locally Few Colors

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Automata, Languages and Programming (ICALP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6755))

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Abstract

In [13], Erdős et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any Δ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than Δ different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring).

We investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given. This generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture, we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2 − 2/(Δ + 1) if the given local coloring is not a biclique coloring. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio of \(2- \Omega(1)\frac{\ln \ln \Delta}{\ln \Delta}\). This matches known inapproximability results for the special case of bounded degree graphs.

Moreover, we show that the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs G P of incomparable pairs defined in the dimension theory of partial orders. We show that G P satisfies properties (i) and (ii) where Δ − 1 is the maximum number of predecessors (or successors) of each job.

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Author information

Authors and Affiliations

  1. Faculty of Informatics, University of Lugano (USI), 6904, Lugano, Switzerland

    Fabian Kuhn

  2. Dalle Molle Institute for Artificial Intelligence (IDSIA), 6928, Manno, Switzerland

    Monaldo Mastrolilli

Authors
  1. Fabian Kuhn
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  2. Monaldo Mastrolilli
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Editor information

Editors and Affiliations

  1. ICE-TCS, School of Computer Science, Reykjavik University, Menntavegur 1, IS 101, Reykjavik, Iceland

    Luca Aceto

  2. Fakultät für Informatik, Universität Wien, Universitätsstraße10/9, 1090, Wien, Österreich, Austria

    Monika Henzinger

  3. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Praha 1, Czech Republic

    Jiří Sgall

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© 2011 Springer-Verlag Berlin Heidelberg

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Kuhn, F., Mastrolilli, M. (2011). Vertex Cover in Graphs with Locally Few Colors. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_42

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  • DOI: https://doi.org/10.1007/978-3-642-22006-7_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22005-0

  • Online ISBN: 978-3-642-22006-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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