Advertisement

Constraints

, Volume 21, Issue 3, pp 375–393 | Cite as

Computing the Ramsey number R(4,3,3) using abstraction and symmetry breaking

  • Michael Codish
  • Michael Frank
  • Avraham Itzhakov
  • Alice Miller
Article

Abstract

The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for almost 50 years. This paper presents a methodology based on abstraction and symmetry breaking that applies to solve hard graph edge-coloring problems. The utility of this methodology is demonstrated by using it to compute the value R(4, 3, 3) = 30. Along the way it is required to first compute the previously unknown set \(\mathcal {R}(3,3,3;13)\) consisting of 78,892 Ramsey colorings.

Keywords

Ramsey numbers SAT solving Symmetry breaking 

Notes

Acknowledgments

We thank Stanislaw Radziszowski for his guidance and comments which helped improve the presentation of this paper. In particular Stanislaw proposed to show that our technique is able to find the (4, 3, 3; 29) coloring depicted as Fig. 4.

References

  1. 1.
    Ahmed, T. (2011). On computation of exact ven der Waerden numbers. Integers, 11.Google Scholar
  2. 2.
    Appel, K., & Haken, W. (1976). Every map is four colourable. Bulletin of the American Mathematical Society, 82, 711–712.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Codish, M., Miller, A., Prosser, P., & Stuckey, P.J. (2013). Breaking symmetries in graph representation. In Rossi, F. (Ed.), Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, China. IJCAI/AAAI. http://www.aaai.org/ocs/index.php/IJCAI/IJCAI13/paper/view/6480.
  4. 4.
    Codish, M., Miller, A., Prosser, P., & Stuckey, P.J. (2014). Constraints for symmetry breaking in graph representation. Full version of [3] (in preparation).Google Scholar
  5. 5.
    Dransfeld, M.R., Liu, L., Marek, V., & Truszczyński, M. (2004). Satisfiability and computing van der Waerden numbers. The Electronic Journal of Combinatorics, 11.Google Scholar
  6. 6.
    Erdös, P., & Gallai, T. (1960). Graphs with prescribed degrees of vertices (in Hungarian). Matematicas Lapok (pp. 264–274). Available from http://www.renyi.hu/p_erdos/1961-05.pdf.
  7. 7.
    Fettes, S.E., Kramer, R.L., & Radziszowski, S.P. (2004). An upper bound of 62 on the classical Ramsey number r(3, 3, 3, 3). Ars Combinatorics, 72.Google Scholar
  8. 8.
    Herwig, P.R., Van Lambalgen, P.M., & Van Maaren, H. (2004). A new method to construct lower bounds for van der Waerden numbers. The Electronic Journal of Combinatorics, 14.Google Scholar
  9. 9.
    Heule, M. Jr., Warren A. Hunt, Jr., & Wetzler, N. (2014). Bridging the gap between easy generation and efficient verification of unsatisfiability proofs. Software Testing, Verification Reliability, 24(8), 593–607. doi: 10.1002/stvr.1549.CrossRefGoogle Scholar
  10. 10.
    Kalbfleisch, J.G. (1966). Chromatic graphs and Ramsey’s theorem. Ph.D. thesis, University of Waterloo.Google Scholar
  11. 11.
    Kouril, M. (2012). Computing the van der Waerden number w(3, 4) = 293. Integers, 12.Google Scholar
  12. 12.
    McKay, B. (1990). Nauty user’s guide (version 1.5). Technical Report TR-CS-90-02, Australian National University, Computer Science Department.Google Scholar
  13. 13.
    McKay, B.D., & Radziszowski, S.P. (1995). R(4, 5) = 25. Journal of Graph Theory, 19(3), 309–322. doi: 10.1002/jgt.3190190304.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    McMillan, K.L. (2002). Applying SAT methods in unbounded symbolic model checking. In Brinksma, E. & Larsen, K.G. (Ed.), Computer Aided Verification, 14th International Conference, Proceedings, Lecture Notes in Computer Science (Vol. 2404, pp. 250–264). Springer.  10.1007/3-540-45657-0_19.
  15. 15.
    Metodi, A., Codish, M., & Stuckey, P. J. (2013). Boolean equi-propagation for concise and efficient SAT encodings of combinatorial problems. Journal of Artificial Intelligence Research (JAIR), 46, 303–341.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Miller, A., & Prosser, P. (2012). Diamond-free degree sequences. Acta Universitatis Sapientiae Informatica, Scienta Publishing House, 4(2), 189–200.zbMATHGoogle Scholar
  17. 17.
    Piwakowski, K. (1997). On Ramsey number r(4, 3, 3) and triangle-free edge-chromatic graphs in three colors. Discrete Mathematics, 164(1-3), 243–249. doi: 10.1016/S0012-365X(96)00057-X.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Piwakowski, K., & Radziszowski, S.P. (1998). 30r(3,3,4)31. Journal of Combinatorial Mathematics and Combinatorial Computing, 27, 135–141.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Piwakowski, K., & Radziszowski, S.P. (2001). Towards the exact value of the Ramsey number R(3, 3, 4). In Proceedings of the 33rd Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Vol. 148, pp. 161–167). Congressus Numerantium. http://www.cs.rit.edu/spr/PUBL/paper44.pdf.
  20. 20.
    Radziszowski, S.P. (2015). Personal communication January.Google Scholar
  21. 21.
    Radziszowski, S.P. (2014). Small Ramsey numbers. Electronic Journal of Combinatorics (1994). http://www.combinatorics.org/ Revision #14: January.
  22. 22.
    Soos, M. (2010). CryptoMiniSAT, v2, 5, 1. http://www.msoos.org/cryptominisat2.Google Scholar
  23. 23.
    Stolee, D. Canonical labelings with nauty. Computational Combinatorics (Blog) (2012). http://computationalcombinatorics.wordpress.com (viewed October 2015).
  24. 24.
    Wetzler, N., Heule, M. Jr., & Warren A. Hunt, Jr. (2014). Drat-trim: Efficient checking and trimming using expressive clausal proofs. In Sinz, C. & Egly, U. (Ed.), Theory and Applications of Satisfiability Testing, 17th International Conference, Proceedings, Lecture Notes in Computer Science (Vol. 8561, pp. 422–429). Springer.doi: 10.1007/978-3-319-09284-3_31.
  25. 25.
    Xu, X., & Radziszowski, S.P. (2015). On some open questions for Ramsey and Folkman numbers. Graph Theory, Favorite Conjectures and Open Problems. (to appear).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Michael Codish
    • 1
  • Michael Frank
    • 1
  • Avraham Itzhakov
    • 1
  • Alice Miller
    • 2
  1. 1.Department of Computer ScienceBen-Gurion University of the NegevBe’er ShevaIsrael
  2. 2.School of Computing ScienceUniversity of GlasgowGlasgowScotland

Personalised recommendations