Vertex Cover in Graphs with Locally Few Colors

  • Fabian Kuhn
  • Monaldo Mastrolilli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


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.


approximation local coloring scheduling vertex cover 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Fabian Kuhn
    • 1
  • Monaldo Mastrolilli
    • 2
  1. 1.Faculty of InformaticsUniversity of Lugano (USI)LuganoSwitzerland
  2. 2.Dalle Molle Institute for Artificial Intelligence (IDSIA)MannoSwitzerland

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