Abstract
Many constraint problems exhibit dominance relations which can be exploited for dramatic reductions in search space. Dominance relations are a generalization of symmetry and conditional symmetry. However, unlike symmetry breaking which is relatively well studied, dominance breaking techniques are not very well understood and are not commonly applied. In this paper, we present formal definitions of dominance breaking, and a generic method for identifying and exploiting dominance relations via dominance breaking constraints. We also give a generic proof of the correctness and compatibility of symmetry breaking constraints, conditional symmetry breaking constraints and dominance breaking constraints.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abdennadher, S., & Schlenker, H. (1999). Proceedings of the Innovative Applications iof Artificial Intelligence Conference, 838–843.
Aldowaisan, T.A. (2001). A new heuristic and dominance relations for no-wait flowshops with setups. Computers & OR, 28(6), 563–584.
Backofen, R., & Will, S. (1999). Excluding symmetries in constraint-based search. Proceedings of the 5th International Conference on Principles and Practice of Constraint Programming CP1999, volume 1713 of LNCS, (pp. 73–87): Springer.
Brucker, P., Drexl, A., Möhring, R.H., Neumann, K., Pesch, E. (1999). Resource-constrained project scheduling, Notation, classification, models, and methods. European Journal of Operational Research, 112(1), 3–41.
Chu, G., & Stuckey, P.J. (2012). A generic method for systematically identifying and exploiting dominance relations. In Proceedings of the 18th International Conference on Principles and Practice of Constraints Programming CP2012, number 7514 in LNCS, (pp. 6–22): Springer.
Chu, G., de la Banda, M.G., Stuckey, P.J. (2010). Automatically exploiting subproblem equivalence in constraint programming In Lodi, A., Milano, M., Toth, P. (Eds.), Proceedings of the 7th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, volume 6140 of LNCS, (pp. 71–86): Springer.
Stuckey, P.J. (2009). Minimizing the maximum number of open stacks by customer search In I.P. Gent (Ed.), Proceedings of the 15th International Conference on Principles and Practice of Constraints Programming, volume 5732 of LNCS, (pp. 242–257): Springer.
Chu, G., & Stuckey, P.J. (2013). Dominance driven search In Schulte, C. (Ed.), Proceedings of the 19th International Conference on Principles and Practice of Constraint Programming, volume 8124 of LNCS, (pp. 217–229): Springer.
Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A. (1996). Symmetry-breaking predicates for search problems. In Proceedings of the 5th International Conference on Principles of Knowledge Representation and Reasoning, (pp. 148–159): Morgan Kaufmann.
Fahle, T., Schamberger, S., Sellmann, M. (2001) In Walsh, T. (Ed.), Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming, volume 2239 of LNCS, (pp. 93–107): Springer.
Feydy, T., & Stuckey, P.J. (2009). Lazy clause generation reengineered. Proceedings of the 15th International Conference on Principles and Practice of Constraints Programming CP2009, volume 5732 of LNCS, (pp. 352–366): Springer.
Flener, P., Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T. (2002). Breaking Row and Column Symmetries in Matrix Models In Hentenryck, P.V. (Ed.), Proceedings of the 8th International Conference on Principles and Practice of Constraints Programming CP2002, volume 2470 of LNCS, (pp. 462–476): Springer.
Flener, P., Pearson, J., Sellmann, M., Van Hentenryck, P. (2006). Static and dynamic structural symmetry breaking. Proceedings of the 12th International Conference on Principles and Practice of Constraints Programming CP2006, (pp. 695–699): Springer.
Filippo F., & Michela M. (2001). Global cut framework for removing symmetries In Walsh, T. (Ed.), Proceedings of the 7th International Conference on Principles and Practice of Constraint Programming, volume 2239 of LNCS, (pp. 77–92): Springer.
de la Banda, G.M. , Stuckey, J.P., Geoffrey, C. (2011). Solving talent scheduling with dynamic programming. INFORMS Journal on Computing, 23(1), 120–137.
Gargani, A., & Refalo, P. (2007). An efficient model and strategy for the steel mill slab design problem. Proc. of CP 2007, volume 4741 of LNCS, (pp. 77–89): Springer.
Gent, I.P., Kelsey, T., Linton, S., McDonald, I., Miguel, I., Smith, B.M. (2005). Conditional symmetry breaking. In Proceedings of the 11th International Conference on Principles and Practice of Constraint Programming CP2005, volume 3709 of LNCS, (pp. 256–270): Springer.
Gent, I.P., & Smith. B.M. (2000). In Proceedings of the European Conference on Artificial Intelligence ECAI2000, IOS Press, 599–603.
Getoor, L., Ottosson, G., Fromherz, M.P.J., Carlson, B. (1997). Effective redundant constraints for online scheduling. In Proceedings of the 14th National Conference on Artificial Intelligence and 9th Innovative Applications of Artificial Intelligence Conference, (pp. 302–307).
Heller, D., Panda, A., Sellmann, M., Yip, J. (2008). Model restarts for structural symmetry breaking. In Proceedings of the 14th International Conference on Principles and Practice of Constraint Programming, (pp. 539–544): Springer.
Hoffman, K.L., & Padberg, M. (1991). Improving LP-representations of zero-one linear programs for branch-and-cut. INFORMS Journal on Computing, 3(2), 121–134.
Toshihide, I. (1977). The power of dominance relations in branch-and-bound algorithms. Journal of the ACM, 24(2), 264–279.
Kalagnanam, J. , Dawande, M., Trumbo, M., Lee. H.S. (1998). Inventory matching problems in the steel industry. Technical report, IBM Research Report, T.J. Watson Research Center, 1998 RC, (p. 21171).
Richard, E. (2004). Korf. Optimal rectangle packing: New results. In Proceedings of the Fourteenth International Conference on Automated Planning and Scheduling (ICAPS 2004), (pp. 142–149).
Roland M. (2005). The challenge of exploiting weak symmetries. In Proc. of the International Workshop on Constraint Solving and Constraint Logic Programming, volume 3978 of LNCS, (pp. 149–163): Springer.
Mears, C., & Garcia, M. (2009). de la Banda, and Mark Wallace. On implementing symmetry detection. Constraints, 14(4), 443–477.
Miller, H.E., Pierskalla, W.P., Rath, G.J. (1976). Nurse scheduling using mathematical programming. Operations Research, 857–870.
Monette, J.N., Schaus, P., Zampelli, S., Deville, Y., Dupont, P. (2007). A CP Approach to the Balanced Academic Curriculum Problem. In Seventh International Workshop on Symmetry and Constraint Satisfaction Problems, volume 7. http://www.info.ucl.ac.be/yde/Papers/SymCon2007_bacp.pdf.
Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., MiniZinc, G.T. (2007). Towards a Standard CP Modelling Language. In Proceedings of the 13th International Conference on Principles and Practice of Constraint Programming CP2007, volume 4741 of LNCS, (pp. 529–543): Springer.
Ohrimenko, O., Stuckey, P.J., Codish, M. (2009). Propagation via lazy clause generation. Constraints, 14(3), 357–391.
Pesant, G. (2004). A regular language membership constraint for finite sequences of variables In Mark Wallace (Ed.), Proceedings of the 10th International Conference on Principles and Practice of Constraint Programming CP2004, volume 3258 of LNCS, (pp. 482–495): Springer.
Prestwich, S., & Beck, J.C. (2004). Exploiting dominance in three symmetric problems Fourth International Workshop on Symmetry and Constraint Satisfaction Problems, (pp. 63–70). http://zeynep.web.cs.unibo.it/SymCon04/SymCon04.pdf.
Proll, L.G., & Smith, B. (1998). Integer linear programming and constraint programming approaches to a template design problem. INFORMS Journal on Computing, 10(3), 265–275.
PSPLib. project scheduling problem library Accessed on 1 March 2012. http://129.187.106.231/psplib.
Puget, J.-F. (2005). Automatic detection of variable and value symmetries. In Proceedings of the 11th International Conference on Principles and Practice of Constraint Programming CP2005, volume 3709 of LNCS, (pp. 475–489): Springer.
Rendl, A. (2010). Effective compilation of constraint models. PhD thesis: St Andrews University. http://hdl.handle.net/10023/973.
Schulte, C., & Stuckey, P.J. (2008). Efficient constraint propagation engines. ACM Transactions on Programming Languages and Systems, 31(1).
Yu, C.F., & Wah, B.W. (1988). Learning dominance relations in combinatorial search problems. IEEE Transactions Software Engineering, 14(8), 1155–1175.
Author information
Authors and Affiliations
Corresponding author
Additional information
An earlier version of this paper was published in 2012 [5]
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Chu, G., Stuckey, P.J. Dominance breaking constraints. Constraints 20, 155–182 (2015). https://doi.org/10.1007/s10601-014-9173-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10601-014-9173-7