Symmetry Breaking in Numeric Constraint Problems

  • Alexandre Goldsztejn
  • Christophe Jermann
  • Vicente Ruiz de Angulo
  • Carme Torras
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6876)

Abstract

Symmetry-breaking constraints in the form of inequalities between variables have been proposed for a few kind of solution symmetries in numeric CSPs. We show that, for the variable symmetries among those, the proposed inequalities are but a specific case of a relaxation of the well-known \(\textsc{lex}\) constraints extensively used for discrete CSPs. We discuss the merits of this relaxation and present experimental evidences of its practical interest.

Keywords

Symmetries Numeric constraints Variable symmetries 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cohen, D., Jeavons, P., Jefferson, C., Petrie, K., Smith, B.: Symmetry definitions for constraint satisfaction problems. Constraints 11(2-3), 115–137 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    COPRIN: The inria project COPRIN examples webpage (2011), http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/
  3. 3.
    Costa, A., Liberti, L., Hansen, P.: Formulation symmetries in circle packing. Electronic Notes in Discrete Mathematics 36, 1303–1310 (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-breaking predicates for search problems. In: KR, pp. 148–159 (1996)Google Scholar
  5. 5.
    Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I.: Breaking row and column symmetries in matrix models. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 462–476. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Gasca, R., Valle, C.D., Cejudo, V., Barba, I.: Improving the computational efficiency in symmetrical numeric constraint satisfaction problems. In: Marín, R., Onaindía, E., Bugarín, A., Santos, J. (eds.) CAEPIA 2005. LNCS (LNAI), vol. 4177, pp. 269–279. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Gent, I., Petrie, K., Puget, J.-F.: Symmetry in constraint programming. In: Handbook of Constraint Programming, pp. 329–376. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  8. 8.
    Gent, I.P.: Groups and constraints: Symmetry breaking during search. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 415–430. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Goldsztejn, A., Lebbah, Y., Michel, C., Rueher, M.: Capabilities of constraint programming in safe global optimization. In: International Symposium on Nonlinear Theory and its Applications, pp. 601–604 (2008)Google Scholar
  10. 10.
    Granvilliers, L., Benhamou, F.: Algorithm 852: Realpaver: an interval solver using constraint satisfaction techniques. ACM T. on Mathematical Software 32, 138–156 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ji, X., Ma, F., Zhang, J.: Solving global unconstrained optimization problems by symmetry-breaking. In: 8th IEEE/ACIS International Conference on Computer and Information Science, pp. 107–111 (2009)Google Scholar
  12. 12.
    Margot, F.: Symmetry in integer linear programming. In: 50 Years of Integer Programming 1958-2008, pp. 647–686. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Merlet, J.-P.: Interval analysis for certified numerical solution of problems in robotics. Applied Mathematics and Computer Science 19(3), 399–412 (2009)MATHGoogle Scholar
  14. 14.
    Meseguer, P., Torras, C.: Exploiting symmetries within constraint satisfaction search. Artif. Intell. 129(1-2), 133–163 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Puget, J.F.: Breaking symmetries in all different problems. In: Proc. 19th International Joint Conference on Artificial Intelligence (IJCAI), pp. 272–277 (2005)Google Scholar
  16. 16.
    Puget, J.-F.: Symmetry breaking revisited. Constraints 10(1), 23–46 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ruiz de Angulo, V., Torras, C.: Exploiting single-cycle symmetries in continuous constraint problems. Journal of Artificial Intelligence Research 34, 499–520 (2009)MathSciNetMATHGoogle Scholar
  18. 18.
    Vu, X.H., Schichl, H., Sam-Haroud, D.: Interval propagation and search on directed acyclic graphs for numerical constraint solving. J. Global Optimization 45(4), 499–531 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Walsh, T.: Parameterized complexity results in symmetry breaking. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 4–14. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alexandre Goldsztejn
    • 1
  • Christophe Jermann
    • 1
  • Vicente Ruiz de Angulo
    • 2
  • Carme Torras
    • 2
  1. 1.Université de Nantes/CNRS LINA (UMR-6241)NantesFrance
  2. 2.Institut de Robòtica i Informàtica Industrial (CSIC-UPC)BarcelonaSpain

Personalised recommendations