Abstract
We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DoubleLex and SnakeLex. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DoubleLex and SnakeLexare often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DoubleLex completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DoubleLex and SnakeLex on some benchmark problems.
Supported by ANR UNLOC project, ANR 08-BLAN-0289-01 and the Australian Government’s Department of Broadband, Communications and the Digital Economy and the ARC.
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References
Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Breaking row and column symmetry in matrix models. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, p. 462. Springer, Heidelberg (2002)
Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Matrix Modelling. Technical Report APES-36-2001, APES group, Presented at Formul 2001 (Workshop on Modelling and Problem Formulation), CP 2001 post-conference workshop (2001)
Crawford, J., Ginsberg, M., Luks, G., Roy, A.: Symmetry breaking predicates for search problems. In: Proceedings of 5th International Conference on Knowledge Representation and Reasoning (KR 1996), pp. 148–159 (1996)
Huczynska, S., McKay, P., Miguel, I., Nightingale, P.: Modelling equidistant frequency permutation arrays: An application of constraints to mathematics. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 50–64. Springer, Heidelberg (2009)
Bessiere, C., Hebrard, E., Hnich, B., Walsh, T.: The complexity of global constraints. In: Proceedings of the 19th National Conference on AI. AAAI, Menlo Park (2004)
Bessiere, C., Hebrard, E., Hnich, B., Walsh, T.: The complexity of global constraints. Constraints 12(2), 239–259 (2007)
Carlsson, M., Beldiceanu, N.: Arc-consistency for a chain of lexicographic ordering constraints. Technical report T2002-18, Swedish Institute of Computer Science (2002)
Katsirelos, G., Narodytska, N., Walsh, T.: Breaking Generator Symmetry. In: Proceedings of SymCon 2009 - 9th International Workshop on Symmetry and Constraint Satisfaction Problems, Colocated with CP 2009 (2009)
Puget, J.F.: Breaking symmetries in all different problems. In: Proceedings of 19th IJCAI, International Joint Conference on Artificial Intelligence, pp. 272–277 (2005)
Shlyakhter, I.: Generating effective symmetry-breaking predicates for search problems. Electronic Notes in Discrete Mathematics 9, 19–35 (2001)
Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Symmetry in matrix models. Technical Report APES-30-2001, APES group. Presented at SymCon 2001 (Symmetry in Constraints), CP, post-conference workshop (2001)
Walsh, T.: Breaking Value Symmetry. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 880–887. Springer, Heidelberg (2007)
Puget, J.F.: Breaking all value symmetries in surjection problems. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 490–504. Springer, Heidelberg (2005)
Law, Y., Lee, J.: Global constraints for integer and set value precedence. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 362–376. Springer, Heidelberg (2004)
Walsh, T.: Symmetry breaking using value precedence. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) ECAI 2006, pp. 168–172. IOS Press, Amsterdam (2006)
Grayland, A., Miguel, I., Roney-Dougal, C.: Snake lex: An alternative to double lex. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 391–399. Springer, Heidelberg (2009)
Hnich, B., Prestwich, S., Selensky, E., Smith, B.: Constraint models for the covering test problem. Constraints 11, 199–219 (2006)
Lubiw, A.: Doubly lexical orderings of matrices. SIAM J. on Computing 16, 854–879 (1987)
Frisch, A., Jefferson, C., Miguel, I.: Constraints for breaking more row and column symmetries. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 318–332. Springer, Heidelberg (2003)
Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Multiset ordering constraints. In: Proceedings of 18th IJCAI, International Joint Conference on Artificial Intelligence (2003)
Gent, I., Smith, B.: Symmetry breaking in constraint programming. In: Horn, W. (ed.) Proceedings of ECAI-2000, pp. 599–603. IOS Press, Amsterdam (2000)
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Katsirelos, G., Narodytska, N., Walsh, T. (2010). On the Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry. In: Cohen, D. (eds) Principles and Practice of Constraint Programming – CP 2010. CP 2010. Lecture Notes in Computer Science, vol 6308. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15396-9_26
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DOI: https://doi.org/10.1007/978-3-642-15396-9_26
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