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


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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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).

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

Mathematics Subject Classification

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