Abstract
We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.
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References
Aguirre, A.: How to Use Symmetries in Boolean Constraint Solving. In: Benhamou, F., Colmerauer, A. (eds.) Constraint Logic Programming: Selected Research, pp. 287–306. MIT Press, Cambridge (1992)
Backofen, R., Will, S.: Excluding Symmetries in Constraint-Based Search. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 73–87. Springer, Heidelberg (1999)
Benhamou, B.: Study of symmetry in constraint satisfaction problems. In: Borning, A. (ed.) PPCP 1994. LNCS, vol. 874, pp. 246–254. Springer, Heidelberg (1994)
Benhamou, B., Sais, L.: Theoretical study of symmetries in propositional calculus and applications. In: Kapur, D. (ed.) CADE 1992. LNCS (LNAI), vol. 607, pp. 281–294. Springer, Heidelberg (1992)
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)
Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-Breaking Predicates for Search Problems. In: Proceedings KR 1996, November 1996, pp. 149–159 (1996)
Fahle, T., Schamberger, S., Sellmann, M.: Symmetry Breaking. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 93–239. Springer, Heidelberg (2001)
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)
Freuder, E.C.: Eliminating Interchangeable Values in Constraint Satisfaction Problems. In: Proceedings AAAI 1991, vol. 1, pp. 227–233 (1991)
Gent, I.P., Harvey, W., Kelsey, T., Linton, S.: Generic SBDD using Computational Group Theory. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 333–347. Springer, Heidelberg (2003)
Gent, I.P., Smith, B.M.: Symmetry Breaking During Search in Constraint Programming. In: Horn, W. (ed.) Proceedings ECAI 2000, pp. 599–603 (2000)
Jégou, P.: Decomposition of Domains Based on the Micro-Structure of Finite Constraint-Satisfaction Problems. In: Proceedings AAAI 1993, pp. 731–736 (1993)
McKay, B.: Practical Graph Isomorphism. Congressus Numerantium 30, 45–87 (1981); The software tool NAUTY is available for download from, http://cs.anu.edu.au/~bdm/nauty/+
Meseguer, P., Torras, C.: Exploiting symmetries within constraint satisfaction search. Artificial Intelligence 129, 133–163 (2001)
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., Markov, I.L.: Automatically Exploiting Symmetries in Constraint Programming. In: Faltings, B.V., Petcu, A., Fages, F., Rossi, F. (eds.) CSCLP 2004. LNCS (LNAI), vol. 3419, pp. 98–112. Springer, Heidelberg (2005)
Roy, P., Pachet, F.: Using Symmetry of Global Constraints to Speed up the Resolution of Constraint Satisfaction Problems. In: Workshop on Non Binary Constraints, ECAI 1998 (August 1998)
Walsh, T.: SAT v CSP. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 441–456. Springer, Heidelberg (2000)
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Cohen, D., Jeavons, P., Jefferson, C., Petrie, K.E., Smith, B.M. (2005). Symmetry Definitions for Constraint Satisfaction Problems. In: van Beek, P. (eds) Principles and Practice of Constraint Programming - CP 2005. CP 2005. Lecture Notes in Computer Science, vol 3709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11564751_5
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DOI: https://doi.org/10.1007/11564751_5
Publisher Name: Springer, Berlin, Heidelberg
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