Abstract
We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetry-breaking constraints.
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References
R. Backofen and S. Will, ‘Excluding symmetries in constraint-based search’, Proc. CP’99, 5th Int. Conf. on Principles and Practice of Constraint Programming, ed., J. Jaffar, LNCS 1713, pp. 73–87. Springer-Verlag, (1999).
J. Crawford, ‘A theoretical analysis of reasoning by symmetry in first-order logic’, Proc. AAAI’92 workshop on tractable reasoning, (1992).
J. Crawford, G. Luks, M. Ginsberg, and A. Roy, ‘Symmetry breaking predicates for search problems’, Proc. KR’96, 5th Int. Conf. on Knowledge Representation and Reasoning, pp. 148–159, (1996).
T. Fahle, S. Schamberger, and M. Sellmann, ‘Symmetry breaking’, Proc. CP’01, 7th Int. Conf. on Principles and Practice of Constraint Programming, ed., T. Walsh, LNCS 2239, pp. 93–107. Springer-Verlag, (2001).
P. Flener, A. M. Frisch, B. Hnich, Z. Kiziltan, I. Miguel, J. Pearson, and T. Walsh, ‘Symmetry in matrix models’, Proc. SymCon’01, CP’01 Workshop on Symmetry in Constraint Satisfaction Problems, (2001). Also technical report APES-36-2001 from http://www.dcs.st-and.ac.uk/~apes/reports/apes-36-2001.ps.gz.
P. Flener, A. M. Frisch, B. Hnich, Z. Kiziltan, I. Miguel, and T. Walsh, ‘Matrix modelling’, Proc. Formul’01, CP’01 Workshop on Modelling and Problem Formulation, (2001). Also technical report APES-36-2001 from http://www.dcs.st-and.ac.uk/~apes/reports/apes-36-2001.ps.gz.
F. Focacci and M. Milano, ‘Global cut framework for removing symmetries’, Proc. CP’01, 7th Int. Conf. on Principles and Practice of Constraint Programming, ed., T. Walsh, LNCS 2239, pp. 77–92. Springer-Verlag, (2001).
E. Freuder, ‘Eliminating Interchangeable Values in Constraint Satisfaction Problems’, Proc. AAAI’91, 9th Nat. Conf. on AI, pp. 227–233, (1991).
E. Freuder, ‘Modelling: The final frontier’, Proc. PACLP’99, 1st Int. Conf. on Practical Application of Constraint Technologies and Logic Programming, (1999).
A. M. Frisch, B. Hnich, Z. Kiziltan, I. Miguel, and T. Walsh, ‘Global constraints for lexicographic orderings’, Proc. CP’02, 8th Int. Conf. on Principles and Practice of Constraint Programming (to appear), (2002).
I. P. Gent, ‘A symmetry breaking constraint for indistinguishable values’, Proc. SymCon’01, CP’01 workshop on Symmetry in Constraints, (2001).
I. P. Gent and B. M. Smith, ‘Symmetry breaking in constraint programming’, Proc. ECAI’OO, Uth Euro. Conf. on AI, ed., W. Horn, pp. 599–603, IOS, (2000).
Z. Kiziltan and B. Hnich, ‘Symmetry breaking in a rack configuration problem’, Proc. IJCAI’01 Workshop on Modelling and Solving Problems with Constraints, (2001).
P. Meseguer and C. Torras, ‘Exploiting symmetries within constraint satisfaction search’, Artificial Intelligence, 129(1-2):133–163, (2001).
S.D. Prestwich, ‘Balanced incomplete block design as satisfiability’, Proc. 12th Irish Conf. on AI and Cognitive Science, (2001).
J.-F. Puget, ‘On the satisfiability of symmetrical constrained satisfaction problems’, Proc. ISMIS’93, eds., J. Komorowski and Z.W. Ras, LNAI 689, pp. 350–361. Springer-Verlag, (1993).
B.M. Smith, ‘Reducing symmetry in a combinatorial design problem’, Proc. CP-AI-OR’01, 3rd Int. Workshop on Integration of AI and OR Techniques in CP, (2001). Also Research Report 2001.01, School of Computing, University of Leeds.
B.M. Smith, ‘Reducing symmetry in a combinatorial design problem’, Proc. IJCAI’01 workshop on Modelling and Solving Problems with Constraints, pp. 105–112, (2001).
B. M. Smith, S. C. Brailsford, P. M. Hubbard, and H. P. Williams, ‘The progressive party problem: Integer linear programming and constraint programming compared’, Constraints, 1:119–138, (1996).
B. M. Smith and I. P. Gent, ‘Reducing symmetry in matrix models: SBDS vs. constraints’, Proc. SymCon’01, CP’01 workshop on Symmetry in Constraints, (2001).
P. Van Hentenryck, L. Michel, L. Perron, and J.-C. Régin, ‘Constraint programming in OPL’, Proc. PPDP’99, Int. Conf. on Principles and Practice of Declarative Programming, ed., G. Nadathur, LNCS 1703, pp. 97–116. Springer-Verlag, (1999).
P. Van Hentenryck ‘The OPL Optimization Programming Language’, The MIT Press, (1999).
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Flener, P. et al. (2002). Breaking Row and Column Symmetries in Matrix Models. In: Van Hentenryck, P. (eds) Principles and Practice of Constraint Programming - CP 2002. CP 2002. Lecture Notes in Computer Science, vol 2470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46135-3_31
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DOI: https://doi.org/10.1007/3-540-46135-3_31
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