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New filtering for AtMostNValue and its weighted variant: A Lagrangian approach

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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

  1. 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.

    Google Scholar 

  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, W.J., Régin, J.-C., Rousseau, L.-M., & Rueher, M. (2012). Improved filtering for weighted circuit constraints. Constraints, 17(3), 205–233.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bessiere, C., Hebrard, E., Hnich, B., Kiziltan, Z., & Walsh, T. (2006). Filtering algorithms for the nvalue constraint. Constraints, 11(4), 271–293.

    Article  MATH  MathSciNet  Google Scholar 

  5. 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.

    Google Scholar 

  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. (2012). A shortest path-based approach to the multileaf collimator sequencing problem. Discrete Applied Mathematics, 160(1–2), 81–99.

    Article  MATH  MathSciNet  Google Scholar 

  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.

  9. 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.

    Article  MATH  MathSciNet  Google Scholar 

  10. 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.

    Article  MATH  MathSciNet  Google Scholar 

  11. Erlenkotter, D. (1978). A dual-based procedure for uncapacitated facility location. Operations Research, 26(6), 992–1009.

    Article  MATH  MathSciNet  Google Scholar 

  12. 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.

    Article  MATH  MathSciNet  Google Scholar 

  13. Fages, J.-G., Lorca, X., & Rousseau, L.-M. (2014). The salesman and the tree: the importance of search in CP. Constraints, 1–18.

  14. 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.

    Google Scholar 

  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.

  16. Gaspers, S., & Szeider, S. (2011). Kernels for global constraints, CoRR. arXiv: 1104.2541.

  17. 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.

    Google Scholar 

  18. Held, M., & Karp, R.M. (1971). The traveling-salesman problem and minimum spanning trees: part II. Mathematical Programming, 1(1), 6–25.

    Article  MATH  MathSciNet  Google Scholar 

  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.

  20. 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.

    MATH  MathSciNet  Google Scholar 

  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.

  22. Narula, S.C., Ogbu, U.I., & Samuelsson, H.M. (1977). An algorithm for the p-median problem. Operations Research, 25(4), 709–713.

    Article  MATH  MathSciNet  Google Scholar 

  23. Prud’homme, C., Fages, J.-G., & Lorca, X. (2014). Choco3 Documentation. TASC, INRIA Rennes, LINA CNRS UMR 6241, COSLING S.A.S.

  24. 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.

    Google Scholar 

  25. Sellmann, M., & Fahle, T. (2003). Constraint programming based lagrangian relaxation for the automatic recording problem. Annals OR, 118(1–4), 17–33.

    Article  MATH  MathSciNet  Google Scholar 

  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.

  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.

  28. Wolsey, L.A. (1998). Integer programming. Wiley-Interscience series in discrete mathmatics and optimization. New York: Wiley.

    Google Scholar 

  29. 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.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Univ. Grenoble Alpes, G-SCOP, 38000, Grenoble, France

    Hadrien Cambazard

  2. CNRS, G-SCOP, 38000, Grenoble, France

    Hadrien Cambazard

  3. COSLING S.A.S., 44307, Nantes Cedex 3, France

    Jean-Guillaume Fages

Authors
  1. Hadrien Cambazard
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  2. Jean-Guillaume Fages
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Correspondence to Hadrien Cambazard.

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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

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