Advertisement

Constraints

, Volume 19, Issue 3, pp 270–308 | Cite as

Consistency techniques for polytime linear global cost functions in weighted constraint satisfaction

  • J. H. M. Lee
  • K. L. Leung
  • Y. W. Shum
Article

Abstract

Lee and Leung make practical the consistency enforcement of global cost functions in Weighted Constraint Satisfaction Problems (WCSPs). The main idea of their approach lies in the derivation of polynomial time algorithms for the computation of the minimum cost of global cost functions. In this paper, we investigate how soft arc consistency can also be applied on global cost functions with no known efficient minimum cost computation algorithms. We propose polytime linear projection-safe (PLPS) cost functions, which have a polynomial size integer linear formulation and can maintain this good property across projection/extension operations. We observe that the minimum of the linear relaxation gives a good approximation to the minimum of the integer formulation. This is used as the basis for the enforcement of relaxed forms of existing soft arc consistencies. By using the linear formulations, we can easily enforce conjunctions of overlapping PLPS, which give stronger pruning power. We further propose polytime integral linear projection-safe (PILPS) cost functions, which are PLPS cost functions with guaranteed integral solutions to the linear relaxation. We prove theorems to compare the consistency strengths among PLPS, PILPS and their conjunctions. Extensive experimentations are conducted to compare our proposed algorithms against state of the art global cost functions consistency enforcement algorithms and integer programming. Empirical results agree with our theoretical predictions, and confirm orders of magnitude improvement in terms of pruning and runtime by our proposals.

Keywords

Constraint optimization Global constraints Linear programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggoun, A., & Beldiceanu, N. (1993). Extending CHIP in order to solve complex scheduling and placement problems. Mathematical and Computer Modelling, 17(7), 57–73.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Akplogan, M., de Givry, S., Métivier, J.-P., Quesnel, G., Joannon, A., Garcia, F. (2013). Solving the crop allocation problem using hard and soft constraints. RAIRO - Operations Research, 47(2), 151–172.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Allouche, D., Bessière, C., Boizumault, P., de Givry, S., Gutierrez, P., Loudni, S., Métivier, J.-P., Schiex, T. (2012). Filtering decomposable global cost functions. In Proceedings of AAAI’12, (pp. 407–413).Google Scholar
  4. 4.
    Allouche, D., Traoré, S., André, I., de Givry, S., Katsirelos, G., Barbe, S., Schiex, T. (2012). Computational protein design as a cost function network optimization problem. In Proceedings of CP’12, (pp. 840–849).Google Scholar
  5. 5.
    Beldiceanu, N., Carlsson, M., Rampon, J.X. (2005). Global constraint catalog. Technical Report T2005- 08, Swedish Institute of Computer Science. http://www.emn.fr/x-info/sdemasse/gccat/.
  6. 6.
    Beldiceanu, N., & Contejean, E. (1994). Introducing global constraints in CHIP. Mathematical and Computer Modelling, 20(12), 97–123.CrossRefMATHGoogle Scholar
  7. 7.
    Beldiceanu, N., Katriel, I., Thiel, S. (2004). Filtering algorithms for the same constraints. In Proceedings of CPAIOR’04, (pp. 65–79).Google Scholar
  8. 8.
    Bessière, C., & Hentenryck, P.V. (2003). To be or not to be … a global constraint. In Proceedings of CP’03, (pp. 789–794).Google Scholar
  9. 9.
    Bessière, C., Katsirelos, G., Narodytska, N., Quimper, C.-G., Walsh, T. (2010). Propagating conjunctions of ALLDIFFERENT constraints. In Proceedings of AAAI’10, (pp. 27–32).Google Scholar
  10. 10.
    Bessière, C., & Régin, J.-C. (1997). Arc consistency for general constraint networks: preliminary results. In Proceedings of IJCAI’97, (pp. 398–404).Google Scholar
  11. 11.
    Cabon, B., de Givry, S., Lobjois, L., Schiex, T., Warners, J.P. (1999). Radio link frequency assignment. Constraints, 4(1), 79–89.CrossRefMATHGoogle Scholar
  12. 12.
    Cooper, M.C. (2005). High-order consistency in valued constraint satisfaction. Constraints, 10(3), 283–305.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Cooper, M.C., de Givry, S., Sànchez, M., Schiex, T., Zytnicki, M. (2008). Virtual arc consistency for weighted CSP. In Proceedings of AAAI’08, (pp. 253–258).Google Scholar
  14. 14.
    Cooper, M.C., de Givry, S., Sànchez, M., Schiex, T., Zytnicki, M., Werner, T. (2010). Soft arc consistency revisited. Artificial Intelligence, 174(7–8), 449–478.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Cooper, M.C., de Givry, S., Schiex, T. (2007). Optimal soft arc consistency. In Proceedings of IJCAI’07, (pp. 68–73).Google Scholar
  16. 16.
    Cooper, M.C., de Roquemaurel, M., Régnier, P. (2011). A weighted CSP approach to cost-optimal planning. AI Communications, 24(1), 1–29.MATHMathSciNetGoogle Scholar
  17. 17.
    Cooper, M.C., & Schiex, T. (2004). Arc consistency for soft constraints. Artifical Intelligence, 154, 199–227.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dantzig, G.B. (1963). Linear programming and extensions. Princeton University Press.Google Scholar
  19. 19.
    de Givry, S., Heras, F., Zytnicki, M., Larrosa, J. (2005). Existential arc consistency: getting closer to full arc consistency in weighted CSPs. In Proceedings of IJCAI’05, (pp. 84–89).Google Scholar
  20. 20.
    Hooker, J.N. (2007). Integrated methods for optimization. Springer.Google Scholar
  21. 21.
    Katriel, I., & Thiel, S. (2005). Complete bound consistency for the global cardinality constraint. Constraints, 10(3), 115–135.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Koster, A.M. (1999). Frequency assignment: models and algorithms. PhD thesis, University of Maastricht.Google Scholar
  23. 23.
    Larrosa, J. (2002). Node and arc consistency in weighted CSP. In Proceedings of AAAI’02, (pp. 48–53).Google Scholar
  24. 24.
    Larrosa, J. (2003). In the quest of the best form of local consistency for weighted CSP. In Proceedings of IJCAI’03, (pp. 239–244).Google Scholar
  25. 25.
    Larrosa, J., & Schiex, T. (2004). Solving weighted CSP by maintaining arc consistency. Artificial Intelligence, 159(1–2), 1–26.CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Lauriere, J.-L. (1978). A language and a program for stating and solving combinatorial problems. Artificial Intelligence, 10(1), 29–127.CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Lee, J.H.M., & Leung, K.L. (2009). Towards efficient consistency enforcement for global constraints in weighted constraint satisfaction. In Proceedings of IJCAI’09, (pp. 559–565).Google Scholar
  28. 28.
    Lee, J.H.M., & Leung, K.L. (2010). A stronger consistency for soft global constraints in weighted constraint satisfaction. In Proceedings of AAAI’10, (pp. 121–127).Google Scholar
  29. 29.
    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, 257–292.MATHMathSciNetGoogle Scholar
  30. 30.
    Lee, J.H.M., Leung, K.L., Wu, Y. (2012). Polynomially decomposable global cost functions in weighted constraint satisfaction. In Proceedings of AAAI’12, (pp. 507–513).Google Scholar
  31. 31.
    Lee, J.H.M., Leung, K.L., Shum, Y.W. (2012). Propagating polynomially (integral) linear projection-safe global cost functions in WCSPs. In Proceedings of ICTAI’12, (pp. 9–16).Google Scholar
  32. 32.
    Lee, J.H.M., & Shum, Y.W. (2011). Modeling soft global constraints as linear programs in weighted constraint satisfaction. In Proceedings of ICTAI’11, (pp. 305–312).Google Scholar
  33. 33.
    Lenstra, J., Kan, A.R., Brucker, P. (1977). Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1, 343–362.CrossRefGoogle Scholar
  34. 34.
    Maher, M.J., Narodytska, N., Quimper, C.-G., Walsh, T. (2008). Flow-based propagators for the SEQUENCE and related global constraints. In Proceedings of CP’08, (pp. 159–174).Google Scholar
  35. 35.
    Papadimitriou, C. H., & Steiglitz, K. (1982). Combinatorial optimization: Algorithms and complexity. Prentice-Hall.Google Scholar
  36. 36.
    Pearl, J. (1988). Probabilistic reasoning in intelligent systems: Networks of plausible inference. San Mateo: Morgan Kaufmann.Google Scholar
  37. 37.
    Pesant, G. (2004). A regular language membership constraint for finite sequences of variables. In Proceedings of CP’04, (pp. 482–495).Google Scholar
  38. 38.
    Petit, T., Régin, J.-C., Bessière, C. (2001). Specific filtering algorithm for over-constrained problems. In Proceedings of CP’01, (pp. 451–463).Google Scholar
  39. 39.
    Quimper, C.-G., López-Ortiz, A., Beek, P.V., Golynski, A. (2004). Improved algorithms for the global cardinality constraint. In Proceedings of CP’04, (pp. 542–556).Google Scholar
  40. 40.
    Régin, J.-C. (1996). Generalized arc consistency for global cardinality constraints. In Proceedings of AAAI’96, (pp. 209–215).Google Scholar
  41. 41.
    Régin, J.-C. (2005). Combination of among and cardinality constraints. In Proceedings of CPAIOR’05, (pp. 288–303).Google Scholar
  42. 42.
    Rossi, F., van Beek, P., Walsh, T. (2006). Handbook of constraint programming. Elsevier.Google Scholar
  43. 43.
    Sànchez, M., de Givry, S., Schiex, T. (2008). Mendelian error detection in complex pedigrees using weighted constraint satisfaction techniques. Constraints, 13(1–2), 130–154.CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Sandholm, T. (1999). An algorithm for optimal winner determination in combinatorial auctions. In Proceedings of IJCAI’99, (pp. 542–547).Google Scholar
  45. 45.
    Schiex, T., Fargier, H., Verfaillie, G. (1995). Valued constraint satisfaction problems: hard and easy problems. In Proceedings of IJCAI’95, (pp. 631–639).Google Scholar
  46. 46.
    Solnon, C., Cung, V., Nguyen, A., Artigues, C. (2008). The car sequencing problem: overview of state-of-the-art methods and industrial case-study of the ROADDEF’2005 challenge problem. European Journal of Operational Research, 191(3), 912–927.CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    van Hoeve, W.-J., Pesant, G., Rousseau, L.-M. (2006). On global warming: flow-based soft global constraints. Journal of Heuristics, 12(4–5), 347–373.CrossRefMATHGoogle Scholar
  48. 48.
    Wolsey, L. (1998). Integer programming. Wiley.Google Scholar
  49. 49.
    Zytnicki, M., Gaspin, C., Schiex, T. (2009). Bounds arc consistency for weighted CSPs. Journal of Artificial Intelligence Research, 35, 593–621.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Chinese University of Hong KongShatinHong Kong

Personalised recommendations