Advertisement

New Integer Linear Programming Models for the Vertex Coloring Problem

  • Adalat Jabrayilov
  • Petra Mutzel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

The vertex coloring problem asks for the minimum number of colors that can be assigned to the vertices of a given graph such that each two neighbors have different colors. The problem is NP-hard. Here, we introduce new integer linear programming formulations based on partial-ordering. They have the advantage that they are as simple to work with as the classical assignment formulation, since they can be fed directly into a standard integer linear programming solver. We evaluate our new models using Gurobi and show that our new simple approach is a good alternative to the best state-of-the-art approaches for the vertex coloring problem. In our computational experiments, we compare our formulations with the classical assignment formulation and the representatives formulation on a large set of benchmark graphs as well as randomly generated graphs of varying size and density. The evaluation shows that the partial-ordering based models dominate both formulations for sparse graphs, while the representatives formulation is the best for dense graphs.

Keywords

Graph coloring Vertex coloring Integer linear programming 

Notes

Acknowledgements

This work was partially supported by DFG, RTG 1855.

References

  1. 1.
    Achterberg, T., Berthold, T., Koch, T., Wolter, K.: Constraint integer programming: a new approach to integrate CP and MIP. In: Perron, L., Trick, M.A. (eds.) CPAIOR 2008. LNCS, vol. 5015, pp. 6–20. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-68155-7_4 CrossRefGoogle Scholar
  2. 2.
    Burke, E.K., Mareček, J., Parkes, A.J., Rudová, H.: A supernodal formulation of vertex colouring with applications in course timetabling. Ann. Oper. Res. 179(1), 105–130 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Campêlo, M.B., Campos, V.A., Corrêa, R.C.: On the asymmetric representatives formulation for the vertex coloring problem. Discrete Appl. Math. 156(7), 1097–1111 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Campêlo, M.B., Corrêa, R.C., Frota, Y.: Cliques, holes and the vertex coloring polytope. Inf. Process. Lett. 89(4), 159–164 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Campos, V., Corrêa, R.C., Delle Donne, D., Marenco, J., Wagler, A.: Polyhedral studies of vertex coloring problems: The asymmetric representatives formulation. ArXiv e-prints, August 2015Google Scholar
  6. 6.
    Cornaz, D., Furini, F., Malaguti, E.: Solving vertex coloring problems as maximum weight stable set problems. Disc. Appl. Math. 217(Part 2), 151–162 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Benchmarking machines and testing solutions (2002). http://mat.gsia.cmu.edu/COLOR02/BENCHMARK/benchmark.tar
  8. 8.
    Eppstein, D.: Small maximal independent sets and faster exact graph coloring. J. Graph Algorithms Appl. 7(2), 131–140 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, CA, USA (1979)Google Scholar
  10. 10.
    Gualandi, S. Chiarandini, M.: Graph coloring instances (2017). https://sites.google.com/site/graphcoloring/vertex-coloring
  11. 11.
    Gualandi, S., Malucelli, F.: Exact solution of graph coloring problems via constraint programming and column generation. INFORMS J. Comput. 24(1), 81–100 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hansen, P., Labbé, M., Schindl, D.: Set covering and packing formulations of graph coloring: algorithms and first polyhedral results. Discrete Optim. 6(2), 135–147 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Held, S., Cook, W., Sewell, E.: Maximum-weight stable sets and safe lower bounds for graph coloring. Math. Program. Comput. 4(4), 363–381 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jabrayilov, A., Mallach, S., Mutzel, P., Rüegg, U., von Hanxleden, R.: Compact layered drawings of general directed graphs. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 209–221. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-50106-2_17 CrossRefGoogle Scholar
  15. 15.
  16. 16.
    Johnson, D.S., Trick, M. (eds.): Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, 1993. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. AMS, Providence (1996)Google Scholar
  17. 17.
    Malaguti, E., Monaci, M., Toth, P.: An exact approach for the vertex coloring problem. Discrete Optim. 8(2), 174–190 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Malaguti, E., Toth, P.: A survey on vertex coloring problems. Int. Trans. Oper. Res. 17, 1–34 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mehrotra, A., Trick, M.: A column generation approach for graph coloring. INFORMS J. Comput. 8(4), 344–354 (1996)CrossRefMATHGoogle Scholar
  20. 20.
    Méndez-Díaz, I., Zabala, P.: A branch-and-cut algorithm for graph coloring. Discrete Appl. Math. 154(5), 826–847 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Méndez-Díaz, I., Zabala, P.: A cutting plane algorithm for graph coloring. Discrete Appl. Math. 156(2), 159–179 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Segundo, P.S.: A new DSATUR-based algorithm for exact vertex coloring. Comput. Oper. Res. 39(7), 1724–1733 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sewell, E.: An improved algorithm for exact graph coloring. In: Johnson and Trick [16], pp. 359–373Google Scholar
  24. 24.
    Trick, M.: DIMACS graph coloring instances (2002). http://mat.gsia.cmu.edu/COLOR02/

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceTU Dortmund UniversityDortmundGermany

Personalised recommendations