Abstract
Constraint programs containing a matrix of two (or more) dimensions of decision variables often have row and column symmetries: in any assignment to the variables the rows can be swapped and the columns can be swapped without affecting whether or not the assignment is a solution. This introduces an enormous amount of redundancy when searching a space of partial assignments. It has been shown previously that one can remove consistently some of these symmetries by extending such a program with constraints that require the rows and columns to be lexicographically ordered. This paper identifies and studies the properties of a new additional constraint—the first row is less than or equal to all permutations of all other rows—that can be added consistently to break even more symmetries. Two alternative implementations of this stronger symmetry-breaking method are investigated, one of which employs a new algorithm that in time linear in the size of the matrix enforces the constraint that the first row is less than or equal to all permutations of all other rows. It is demonstrated experimentally that our method for breaking more symmetries substantially reduces search effort.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Schellenberg, P.J., Anderson, B.A., Stinson, D.R.: The existance of Howell designs of even side. J. Combin. Theory, 23–55 (1984)
Carlsson, M., Beldiceanu, N.: Arc-consistency for a chain of lexicographic ordering constraints. Technical Report T2002-18, Swedish Institute of Computer Science (2002)
Conway, J.H., Slone, N.J.A.: Sphere-Packings, Lattices and Groups, 3rd edn. Springer, Heidelberg (1992)
Crawford, J., Ginsberg, M.L., Luck, E., Roy, A.: Symmetry-breaking predicates for search problems. In: Proceedings of KR 1996: Principles of Knowledge Representation and Reasoning, pp. 148–159 (1996)
Flener, P., Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Breaking row and column symmetries in matrix models. In: Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming, pp. 462–476 (2002)
Flener, P., Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Matrix modelling: Exploiting common patterns in constraint programming. In: Proceedings of the International Workshop on Reformulating Constraint Satisfaction Problems, pp. 27–41 (2002)
Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Global constraints for lexicographic orderings. In: Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming, pp. 93–108 (2002)
Frisch, A.M., Miguel, I., Kiziltan, Z., Hnich, B., Walsh, T.: Multiset ordering constraints. In: Proceedings of the 18th International Joint Conference on AI (2003)
Guy, R.K.: Unsolved Problems in Number Theory. Springer, Heidelberg (1994)
Hammons, A.R., Kumar, P.V., Slone, N.J.A., Solé, P.: The Z4-linearity of kerdock, goethals and related codes. IEEE Trans. Information Theory (1993)
Molnar, A.E.: A matrix problem. American Mathematical Monthly 81, 383–384 (1974)
Mullin, R.C., Stanton, R.G.: Group matrices and balanced weighing designs. Utilitas Math. 8, 277–301 (1975)
van Rees, G.H.J., Schellenberg, P.J., Vanstone, S.A.: Four pairwise orthogonal latin squares of order 15. Ars Combin. 6, 141–150 (1978)
Shlyakhter, I.: Generating effective symmetry-breaking predicates for search problems. Discrete Applied Mathematics (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Frisch, A.M., Jefferson, C., Miguel, I. (2003). Constraints for Breaking More Row and Column Symmetries. In: Rossi, F. (eds) Principles and Practice of Constraint Programming – CP 2003. CP 2003. Lecture Notes in Computer Science, vol 2833. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45193-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-540-45193-8_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20202-8
Online ISBN: 978-3-540-45193-8
eBook Packages: Springer Book Archive