Clause Learning and New Bounds for Graph Coloring

  • Emmanuel HebrardEmail author
  • George KatsirelosEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11008)


Graph coloring is a major component of numerous allocation and scheduling problems.

We introduce a hybrid CP/SAT approach to graph coloring based on exploring Zykov’s tree: for two non-neighbors, either they take a different color and there might as well be an edge between them, or they take the same color and we might as well merge them. Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph.

We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; and a branching heuristic emulating Brelaz on the Zykov tree.

The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds.


  1. 1.
    Aardal, K.I., Van Hoesel, S.P.M., Koster, A.M.C.A., Mannino, C., Sassano, A.: Models and solution techniques for frequency assignment problems. Ann. Oper. Res. 153(1), 79–129 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brélaz, D.: New methods to color the vertices of a graph. Commun. ACM 22(4), 251–256 (1979)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chaitin, G.J., Auslander, M.A., Chandra, A.K., Cocke, J., Hopkins, M.E., Markstein, P.W.: Register allocation via coloring. Comput. Lang. 6(1), 47–57 (1981)CrossRefGoogle Scholar
  4. 4.
    De Cat, B., Denecker, M., Bruynooghe, M., Stuckey, P.: Lazy model expansion: interleaving grounding with search. J. Artif. Intell. Res. 52, 235–286 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Eppstein, D., Löffler, M., Strash, D.: Listing all maximal cliques in large sparse real-world graphs. ACM J. Exp. Algorithmics 18(3.1–3.21) (2013)CrossRefGoogle Scholar
  6. 6.
    Frost, D., Dechter, R.: Look-ahead value ordering for constraint satisfaction problems. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI-1995), pp. 572–578 (1995)Google Scholar
  7. 7.
    Gomes, C., Selman, B., Kautz, H.: Boosting combinatorial search through randomization. In: Proceedings of the 15th National Conference on Artificial Intelligence (AAAI-1998), pp. 431–438 (1998)Google Scholar
  8. 8.
    Huang, J.: The effect of restarts on the efficiency of clause learning. In: Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI-2007) (2007)Google Scholar
  9. 9.
    Jiang, H., Li, C.-M., Manyà, F.: An exact algorithm for the maximum weight clique problem in large graphs. In: Proceedings of the 31st Conference on Artificial Intelligence (AAAI-2017), pp. 830–838 (2017)Google Scholar
  10. 10.
    Katsirelos, G., Bacchus, F.: Generalized nogoods in CSPs. In: Proceedings of the 20th National Conference on Artificial Intelligence (AAAI-2005), pp. 390–396 (2005)Google Scholar
  11. 11.
    Lick, D.R., White, A.T.: k-degenerate graphs. Can. J. Math. 22, 1082–1096 (1970)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lin, J., Cai, S., Luo, C., Su, K.: A reduction based method for coloring very large graphs. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI-2017), pp. 517–523 (2017)Google Scholar
  13. 13.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP–a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mehrotra, A., Trick, M.A.: A column generation approach for graph coloring. INFORMS J. Comput. 8(4), 344–354 (1996)CrossRefGoogle Scholar
  15. 15.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 39th Design Automation Conference (DAC-2001), pp. 530–535 (2001)Google Scholar
  16. 16.
    Mycielski, J.: Sur le coloriage des graphes. Colloq. Math 3, 161–162 (1955)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ohrimenko, O., Stuckey, P.J., Codish, M.: Propagation = lazy clause generation. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 544–558. Springer, Heidelberg (2007). Scholar
  18. 18.
    Park, T., Lee, C.Y.: Application of the graph coloring algorithm to the frequency assignment problem. J. Oper. Res. Soc. Jpn. 39(2), 258–265 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Schaafsma, B., Heule, M.J.H., van Maaren, H.: Dynamic symmetry breaking by simulating zykov contraction. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 223–236. Springer, Heidelberg (2009). Scholar
  20. 20.
    Stergiou, K.: Heuristics for dynamically adapting propagation. In: Proceedings of the 18th European Conference on Artificial Intelligence (ECAI-2008), pp. 485–489 (2008)Google Scholar
  21. 21.
    Trick, M.A. (ed.): Computational Symposium on Graph Coloring and its Generalizations (COLOR-2002) (2002)Google Scholar
  22. 22.
    Van Gelder, A.: Another look at graph coloring via propositional satisfiability. Discrete Appl. Math. 156(2), 230–243 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Van Hentenryck, P., Ågren, M., Flener, P., Pearson, J.: Tractable symmetry breaking for CSPs with interchangeable values. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI-2003), pp. 277–282 (2003)Google Scholar
  24. 24.
    Walsh, T.: Breaking value symmetry. In: Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI-2008), pp. 880–887 (2008)Google Scholar
  25. 25.
    Zhou, Z., Li, C.-M., Huang, C., Ruchu, X.: An exact algorithm with learning for the graph coloring problem. Comput. Oper. Res. 51, 282–301 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zykov, A.A.: On some properties of linear complexes. Mat. Sb. (N.S.) 24(66)(2), 163–188 (1949). Scholar

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Authors and Affiliations

  1. 1.LAAS-CNRS, Université de Toulouse, CNRSToulouseFrance
  2. 2.MIAT, UR-875, INRAToulouseFrance

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