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Maximum-weight stable sets and safe lower bounds for graph coloring

Mathematical Programming Computation volume 4, pages 363–381 (2012)Cite this article

Abstract

The best method known for determining lower bounds on the vertex coloring number of a graph is the linear-programming column-generation technique, where variables correspond to stable sets, first employed by Mehrotra and Trick in 1996. We present an implementation of the method that provides numerically-safe results, independent of the floating-point accuracy of linear-programming software. Our work includes an improved branch-and-bound algorithm for maximum-weight stable sets and a parallel branch-and-price framework for graph coloring. Computational results are presented on a collection of standard test instances, including the unsolved challenge problems created by David S. Johnson in 1989.

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Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Lennéstr. 2, 53113, Bonn, Germany

    Stephan Held

  2. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA, 30332-0205, USA

    William Cook

  3. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026, USA

    Edward C. Sewell

Authors
  1. Stephan Held
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  2. William Cook
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  3. Edward C. Sewell
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Corresponding author

Correspondence to Stephan Held.

Additional information

Stephan Held’s research was supported by a postdoctoral fellowship grant from the DAAD. William Cook’s research was supported by NSF Grant CMMI-0726370 and ONR Grant N00014-12-1-0030.

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Held, S., Cook, W. & Sewell, E.C. Maximum-weight stable sets and safe lower bounds for graph coloring. Math. Prog. Comp. 4, 363–381 (2012). https://doi.org/10.1007/s12532-012-0042-3

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  • DOI: https://doi.org/10.1007/s12532-012-0042-3

Keywords

  • Graph coloring
  • Fractional chromatic number
  • Column generation
  • Maximum-weight stable set
  • Safe computations

Mathematics Subject Classification

  • 90-04
  • 90-08
  • 90C27
  • 90C35