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New filtering for AtMostNValue and its weighted variant: a Lagrangian approach. (English) Zbl 1327.90130
Summary: The AtMostNValue global constraint, which restricts the maximum number of distinct values taken by a set of variables, is a well known NP-Hard global constraint. The weighted version of the constraint, AtMostWValue, where each value is associated with a weight or cost, is a useful and natural extension. Both constraints occur in many industrial applications where the number and the cost of some resources have to be minimized. This paper introduces a new filtering algorithm based on a Lagrangian relaxation for both constraints. This contribution is illustrated on problems related to facility location, which is a fundamental class of problems in operations research and management sciences. Preliminary evaluations show that the filtering power of the Lagrangian relaxation can provide significant improvements over the state-of-the-art algorithm for these constraints. We believe it can help to bridge the gap between constraint programming and linear programming approaches for a large class of problems related to facility location.

MSC:
 90C11 Mixed integer programming
Software:
AtMostNValue; ToulBar2
Full Text:
References:
 [1] Beldiceanu, N; Carlsson, M; Walsh, T (ed.), Pruning for the minimum constraint family and for the number of distinct values constraint family, (2001), Berlin Heidelberg · Zbl 1067.68611 [2] Beldiceanu, N., Carlsson, M., & Thiel, S. (2002). Cost-filtering algorithms for the two sides of the sum of weights of distinct values constraint. Technical report - T2002-14: Swedish Institute of Computer Science. [3] Benchimol, P; Van Hoeve, WJ; Régin, J-C; Rousseau, L-M; Rueher, M, Improved filtering for weighted circuit constraints, Constraints, 17, 205-233, (2012) · Zbl 1309.90115 [4] Bessiere, C; Hebrard, E; Hnich, B; Kiziltan, Z; Walsh, T, Filtering algorithms for the nvalue constraint, Constraints, 11, 271-293, (2006) · Zbl 1114.68064 [5] Bessiere, C; Katsirelos, G; Narodytska, N; Quimper, C-G; Walsh, T; Cohen, D (ed.), Decomposition of the nvalue constraint, (2010), Berlin Heidelberg [6] Cambazard, H. Np-hard contraints involving costs: examples of applications and filtering. In Dixièmes Journées Francophones de Programmation par Contraintes - JFPC. 2014. Exposé invité. [7] Cambazard, H; O’Mahony, E; O’Sullivan, B, A shortest path-based approach to the multileaf collimator sequencing problem, Discrete Applied Mathematics, 160, 81-99, (2012) · Zbl 1235.90125 [8] Cambazard, H., & Penz, B. (2012). A constraint programming approach for the traveling purchaser problem. In Milano, M. (Ed.) Principles and Practice of Constraint Programming - 18th International Conference, CP 2012, Quėbec City, QC, Canada, October 8-12, 2012. Proceedings, volume 7514 of Lecture Notes in Computer Science, (pp. 735-749): Springer. · Zbl 1015.90089 [9] Cooper, MC; de Givry, S; Sanchez, M; Schiex, T; Zytnicki, M; Werner, T, Soft arc consistency revisited, Artificial Intelligence, 174, 449-478, (2010) · Zbl 1213.68580 [10] Van den Bergh, J; Belieën, J; De Bruecker, P; Demeulemeester, E; De Boeck, L, Personnel scheduling: a literature review, European Journal of Operational Research, 226, 367-385, (2013) · Zbl 1292.90001 [11] Erlenkotter, D, A dual-based procedure for uncapacitated facility location, Operations Research, 26, 992-1009, (1978) · Zbl 0422.90053 [12] Fages, J-G; Lapègue, T, Filtering atmostnvalue with difference constraints: application to the shift minimisation personnel task scheduling problem, Artificial Intelligence, 212, 116-133, (2014) · Zbl 1407.90150 [13] Fages, J.-G., Lorca, X., & Rousseau, L.-M. (2014). The salesman and the tree: the importance of search in CP. Constraints, 1-18. · Zbl 1237.68188 [14] Focacci, F; Lodi, A; Milano, M; Jaffar, J (ed.), Cost-based domain filtering, (1999), Berlin Heidelberg [15] Fontaine, D., Michel, L.D., & Van Hentenryck, P. Constraint-based lagrangian relaxation. In O’Sullivan, B. (Ed.) Principles and Practice of Constraint Programming - 20th International Conference, CP 2014, Lyon, France, September 8-12, 2014. Proceedings, volume 8656 of Lecture Notes in Computer Science, (pp. 324-339) (p. 2014). Berlin: Springer. · Zbl 1292.90001 [16] Gaspers, S., & Szeider, S. (2011). Kernels for global constraints, CoRR. arXiv: 1104.2541. · Zbl 1407.90150 [17] Geoffrion, AM; Balinski, ML (ed.), Lagrangean relaxation for integer programming, (1974), Berlin Heidelberg · Zbl 0395.90056 [18] Held, M; Karp, RM, The traveling-salesman problem and minimum spanning trees: part II, Mathematical Programming, 1, 6-25, (1971) · Zbl 0232.90038 [19] Kadioglu, S., Malitsky, Y., Sellmann, M., & Tierney, K. (2010). ISAC - instance-specific algorithm configuration. In ECAI, volume 215 of Frontiers in Artificial Intelligence and Applications, (pp. 751-756): IOS Press. · Zbl 0422.90053 [20] Lee, JHM; Leung, KL, Consistency techniques for flow-based projection-safe global cost functions in weighted constraint satisfaction, Journal of Artificial Intelligence Research, 43, 257-292, (2012) · Zbl 1237.68188 [21] Menana, J., & Demassey, S. (2009). Sequencing and counting with the multicost-regular constraint. In Van Hoeve, W.J., & Hooker, J.N. (Eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 6th International Conference, CPAIOR 2009, Pittsburgh, PA, USA, May 27-31, 2009, Proceedings, volume 5547 of Lecture Notes in Computer Science, (pp. 178-192): Springer. · Zbl 1241.68104 [22] Narula, SC; Ogbu, UI; Samuelsson, HM, An algorithm for the p-Median problem, Operations Research, 25, 709-713, (1977) · Zbl 0372.90096 [23] Prud’homme, C., Fages, J.-G., & Lorca, X. (2014). Choco3 Documentation. TASC, INRIA Rennes, LINA CNRS UMR 6241, COSLING S.A.S. · Zbl 1235.90125 [24] Sellmann, M; Wallace, M (ed.), Theoretical foundations of cp-based Lagrangian relaxation, (2004), Berlin Heidelberg · Zbl 1152.68584 [25] Sellmann, M; Fahle, T, Constraint programming based Lagrangian relaxation for the automatic recording problem, Annals OR, 118, 17-33, (2003) · Zbl 1026.90510 [26] Slusky, M.R., & Van Hoeve, W.J. (2013). A lagrangian relaxation for golomb rulers. In Gomes, C.P., & Sellmann, M. (Eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 10th International Conference, CPAIOR 2013, Yorktown Heights, NY, USA, May 18-22, 2013. Proceedings, volume 7874 of Lecture Notes in Computer Science, (pp. 251-267): Springer. · Zbl 1382.68233 [27] Wah, B.W., & Wu, Z. (1999). The theory of discrete lagrange multipliers for nonlinear discrete optimization. In Jaffar, J. (Ed.) Principles and Practice of Constraint Programming - CP’99, 5th International Conference, Alexandria, Virginia, USA, October 11-14, 1999, Proceedings, volume 1713 of Lecture Notes in Computer Science, (pp. 28-42): Springer. · Zbl 0965.90051 [28] Wolsey, L.A. (1998). Integer programming. Wiley-Interscience series in discrete mathmatics and optimization. New York: Wiley. [29] Zhao, X; Luh, PB, New bundle methods for solving Lagrangian relaxation dual problems, Journal of Optimization Theory and Applications, 113, 373-397, (2002) · Zbl 1015.90089
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