## Abstract

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.

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## References

Beldiceanu, N., & Carlsson, M. (2001). Pruning for the minimum constraint family and for the number of distinct values constraint family. In Walsh, T. (Ed.)

*Principles and Practice of Constraint Programming – CP 2001*, volume 2239 of*Lecture Notes in Computer Science*, (pp. 211–224). Berlin Heidelberg: Springer.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.Benchimol, P., Van Hoeve, W.J., Régin, J.-C., Rousseau, L.-M., & Rueher, M. (2012). Improved filtering for weighted circuit constraints.

*Constraints*,*17*(3), 205–233.Bessiere, C., Hebrard, E., Hnich, B., Kiziltan, Z., & Walsh, T. (2006). Filtering algorithms for the nvalue constraint.

*Constraints*,*11*(4), 271–293.Bessiere, C., Katsirelos, G., Narodytska, N., Quimper, C.-G., & Walsh, T. (2010). Decomposition of the nvalue constraint In Cohen, D. (Ed.)

*Principles and Practice of Constraint Programming – CP 2010*, volume 6308 of*Lecture Notes in Computer Science*, (pp. 114–128). Berlin Heidelberg: Springer.Cambazard, H. Np-hard contraints involving costs: examples of applications and filtering. In

*Dixièmes Journées Francophones de Programmation par Contraintes – JFPC.*2014. Exposé invité.Cambazard, H., O’Mahony, E., & O’Sullivan, B. (2012). A shortest path-based approach to the multileaf collimator sequencing problem.

*Discrete Applied Mathematics*,*160*(1–2), 81–99.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.Cooper, M.C., de Givry, S., Sanchez, M., Schiex, T., Zytnicki, M., & Werner, T. (2010). Soft arc consistency revisited.

*Artificial Intelligence*,*174*(7–8), 449–478.Van den Bergh, J., Belieën, J., De Bruecker, P., Demeulemeester, E., & De Boeck, L. (2013). Personnel scheduling: a literature review.

*European Journal of Operational Research*,*226*(3), 367–385.Erlenkotter, D. (1978). A dual-based procedure for uncapacitated facility location.

*Operations Research*,*26*(6), 992–1009.Fages, J.-G., & Lapègue, T. (2014). Filtering atmostnvalue with difference constraints: application to the shift minimisation personnel task scheduling problem.

*Artificial Intelligence*,*212*(0), 116–133.Fages, J.-G., Lorca, X., & Rousseau, L.-M. (2014). The salesman and the tree: the importance of search in CP.

*Constraints*, 1–18.Focacci, F., Lodi, A., & Milano, M. (1999). Cost-based domain filtering In Jaffar, J. (Ed.)

*Principles and Practice of Constraint Programming – CP’99*, volume 1713 of*Lecture Notes in Computer Science*, (pp. 189–203). Berlin Heidelberg: Springer.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.Gaspers, S., & Szeider, S. (2011). Kernels for global constraints,

*CoRR*. arXiv: 1104.2541.Geoffrion, A.M. (1974). Lagrangean relaxation for integer programming. In Balinski, M.L. (Ed.)

*Approaches to integer programming*, volume 2*of mathematical programming studies*, (pp. 82–114). Berlin Heidelberg: Springer.Held, M., & Karp, R.M. (1971). The traveling-salesman problem and minimum spanning trees: part II.

*Mathematical Programming*,*1*(1), 6–25.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.Lee, J.H.M., & Leung, K.L. (2012). Consistency techniques for flow-based projection-safe global cost functions in weighted constraint satisfaction.

*Journal of Artificial Intelligence Research*,*43*(1), 257–292.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.Narula, S.C., Ogbu, U.I., & Samuelsson, H.M. (1977). An algorithm for the p-median problem.

*Operations Research*,*25*(4), 709–713.Prud’homme, C., Fages, J.-G., & Lorca, X. (2014).

*Choco3 Documentation*. TASC, INRIA Rennes, LINA CNRS UMR 6241, COSLING S.A.S.Sellmann, M. (2004). Theoretical foundations of cp-based lagrangian relaxation. In Wallace, M. (Ed.)

*Principles and Practice of Constraint Programming – CP 2004*, volume 3258 of*Lecture Notes in Computer Science*, (pp. 634–647). Berlin Heidelberg: Springer.Sellmann, M., & Fahle, T. (2003). Constraint programming based lagrangian relaxation for the automatic recording problem.

*Annals OR*,*118*(1–4), 17–33.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.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.Wolsey, L.A. (1998).

*Integer programming. Wiley-Interscience series in discrete mathmatics and optimization*. New York: Wiley.Zhao, X., & Luh, P.B. (2002). New bundle methods for solving lagrangian relaxation dual problems.

*Journal of Optimization Theory and Applications*,*113*(2), 373–397.

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Cambazard, H., Fages, JG. New filtering for AtMostNValue and its weighted variant: A Lagrangian approach.
*Constraints* **20**, 362–380 (2015). https://doi.org/10.1007/s10601-015-9191-0

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DOI: https://doi.org/10.1007/s10601-015-9191-0