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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Walsh, T. (ed.): CP 2001. LNCS, vol. 2239. Springer, Heidelberg (2001)
Börger, E., Gräedel, E., Gurevich, Y.: The Classical Decision Problem. Perspect. in Math. Logic. (1997)
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)
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)
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)
Cadoli, M., Schaerf, A.: Compiling problem specifications into SAT. Artif. Intell. (to appear)
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)
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)
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)
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)
Fourer, R., Gay, D.M., Kernigham, B.W.: AMPL: A Modeling Language for Mathematical Programming. Intl. Thomson Publ. (1993)
Hart, W., Istrail, S.: HP Benchmarks, Available at http://www.cs.sandia.gov/tech_reports/compbio/tortilla-hp-benchmarks.html
Köbler, J., Schöning, U., Torán, J.: The graph isomorphism problem: its computational complexity. Birkhauser Press, Basel (1993)
Lau, K.F., Dill, K.A.: A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules 22, 3986–3997 (1989)
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)
McKay, B.D.: Nauty user’s guide, version 2.2 (2003), Available at http://cs.anu.edu.au/~bdm/nauty/nug.pdf
Meseguer, P., Torras, C.: Solving strategies for highly symmetric CSPs. In: Proc. of IJCAI 1999, pp. 400–405. Morgan Kaufmann, San Francisco (1999)
Meseguer, P., Torras, C.: Exploiting symmetries within constraint satisfaction search. Artif. Intell. 129, 133–163 (2001)
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)
Papadimitriou, C.H.: Computational Complexity. Addison Wesley Publ. Co., Reading (1994)
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)
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)
Smith, B.M.: Dual model of permutation problems. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 615–619. Springer, Heidelberg (2001)
Van Hentenryck, P.: The OPL Optimization Programming Language. The MIT Press, Cambridge (1999)
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)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mancini, T., Cadoli, M. (2005). Detecting and Breaking Symmetries by Reasoning on Problem Specifications. In: Zucker, JD., Saitta, L. (eds) Abstraction, Reformulation and Approximation. SARA 2005. Lecture Notes in Computer Science(), vol 3607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11527862_12
Download citation
DOI: https://doi.org/10.1007/11527862_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-27872-6
Online ISBN: 978-3-540-31882-8
eBook Packages: Computer ScienceComputer Science (R0)