Mathematical Programming Computation

, Volume 4, Issue 4, pp 363–381

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

Full Length Paper

DOI: 10.1007/s12532-012-0042-3

Cite this article as:
Held, S., Cook, W. & Sewell, E.C. Math. Prog. Comp. (2012) 4: 363. doi:10.1007/s12532-012-0042-3


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.


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

Mathematics Subject Classification

90-04 90-08 90C27 90C35 

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany
  2. 2.H. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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