Detecting and Breaking Symmetries by Reasoning on Problem Specifications

  • Toni Mancini
  • Marco Cadoli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3607)

Abstract

In this paper we address the problem of detecting and breaking symmetries in combinatorial problems, following the approach of imposing additional symmetry-breaking constraints. Differently from other works in the literature, we attack the problem at the specification level. In fact, many symmetries depend on the structure of the problem, and not on the particular input instance. Hence, they can be easily detected by reasoning on the specification, and appropriate symmetry-breaking formulae generated. We give formal definitions of symmetries and symmetry-breaking formulae on specifications written in existential second-order logic, clarifying the new definitions on some specifications: Graph 3-coloring, Social golfer, and Protein folding problems. Finally, we show experimentally that, applying this technique, even if in a naive way, to specifications written in state-of-the-art languages, e.g., opl, may greatly improve search efficiency.

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References

  1. 1.
    Walsh, T. (ed.): CP 2001. LNCS, vol. 2239. Springer, Heidelberg (2001)MATHGoogle Scholar
  2. 2.
    Börger, E., Gräedel, E., Gurevich, Y.: The Classical Decision Problem. Perspect. in Math. Logic. (1997)Google Scholar
  3. 3.
    Brown, C.A., Finkelstein, L., Purdom, P.W.: Backtrack searching in the presence of symmetry. In: Mora, T. (ed.) AAECC 1988. LNCS, vol. 357, pp. 99–110. Springer, Heidelberg (1989)Google Scholar
  4. 4.
    Cadoli, M., Mancini, T.: Exploiting functional dependencies in declarative problem specifications. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 628–640. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Cadoli, M., Mancini, T.: Using a theorem prover for reasoning on constraint problems. In: Proc. of Intl. W. on Modelling and Reformulating CSPs: Towards Systematisation and Automation, in conj. with CP 2004 (2004)Google Scholar
  6. 6.
    Cadoli, M., Schaerf, A.: Compiling problem specifications into SAT. Artif. Intell. (to appear)Google Scholar
  7. 7.
    Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A.: Symmetry-breaking predicates for search problems. In: Proc. of KR 1996, pp. 148–159. Morgan Kaufmann, San Francisco (1996)Google Scholar
  8. 8.
    Crescenzi, P., Goldman, D., Papadimitriou, C.H., Piccolboni, A., Yannakakis, M.: On the complexity of protein folding. J. of Comp. Biology 5(3), 423–466 (1998)CrossRefGoogle Scholar
  9. 9.
    Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Breaking row and column symmetries in matrix models. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, p. 462. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Focacci, F., Milano, M.: Global cut framework for removing symmetries. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 77–92. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Fourer, R., Gay, D.M., Kernigham, B.W.: AMPL: A Modeling Language for Mathematical Programming. Intl. Thomson Publ. (1993)Google Scholar
  12. 12.
    Hart, W., Istrail, S.: HP Benchmarks, Available at http://www.cs.sandia.gov/tech_reports/compbio/tortilla-hp-benchmarks.html
  13. 13.
    Köbler, J., Schöning, U., Torán, J.: The graph isomorphism problem: its computational complexity. Birkhauser Press, Basel (1993)Google Scholar
  14. 14.
    Lau, K.F., Dill, K.A.: A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules 22, 3986–3997 (1989)CrossRefGoogle Scholar
  15. 15.
    Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S., Scarcello, F.: The DLV System for Knowledge Representation and Reasoning. ACM Trans. on Comp. Logic (to appear)Google Scholar
  16. 16.
    McKay, B.D.: Nauty user’s guide, version 2.2 (2003), Available at http://cs.anu.edu.au/~bdm/nauty/nug.pdf
  17. 17.
    Meseguer, P., Torras, C.: Solving strategies for highly symmetric CSPs. In: Proc. of IJCAI 1999, pp. 400–405. Morgan Kaufmann, San Francisco (1999)Google Scholar
  18. 18.
    Meseguer, P., Torras, C.: Exploiting symmetries within constraint satisfaction search. Artif. Intell. 129, 133–163 (2001)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Annals of Math. and Artif. Intell. 25(3,4), 241–273 (1999)MATHCrossRefGoogle Scholar
  20. 20.
    Papadimitriou, C.H.: Computational Complexity. Addison Wesley Publ. Co., Reading (1994)MATHGoogle Scholar
  21. 21.
    Puget, J.-F.: On the satisfiability of symmetrical constrained satisfaction problems. In: Komorowski, J., Raś, Z.W. (eds.) ISMIS 1993. LNCS, vol. 689, pp. 350–361. Springer, Heidelberg (1993)Google Scholar
  22. 22.
    Ramani, A., Aloul, F.A., Markov, I.L., Sakallak, K.A.: Breaking instance-independent symmetries in exact graph coloring. In: Proc. of DATE 2004, pp. 324–331. IEEE Comp. Society Press, Los Alamitos (2004)Google Scholar
  23. 23.
    Smith, B.M.: Dual model of permutation problems. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 615–619. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  24. 24.
    Van Hentenryck, P.: The OPL Optimization Programming Language. The MIT Press, Cambridge (1999)Google Scholar
  25. 25.
    Van Hentenryck, P., Flener, P., Pearson, J., Ågren, M.: Tractable symmetry breaking for CSPs with interchangeable values. In: Proc. of IJCAI 2003, pp. 277–282. Morgan Kaufmann, San Francisco (2003)Google Scholar
  26. 26.
    Van Hentenryck, P., Flener, P., Pearson, J., Ågren, M.: Compositional derivation of symmetries for constraint satisfaction. TR 2004-022, Dep. of Inf. Tech., Uppsala Univ., Uppsala, Sweden (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Toni Mancini
    • 1
  • Marco Cadoli
    • 1
  1. 1.Dipartimento di Informatica e SistemisticaUniversitá di Roma “La Sapienza” 

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