Advertisement

Consistencies for Ultra-Weak Solutions in Minimax Weighted CSPs Using the Duality Principle

  • Arnaud Lallouet
  • Jimmy H. M. Lee
  • Terrence W. K. Mak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7514)

Abstract

Minimax Weighted Constraint Satisfaction Problems (formerly called Quantified Weighted CSPs) are a framework for modeling soft constrained problems with adversarial conditions. In this paper, we describe novel definitions and implementations of node, arc and full directional arc consistency notions to help reduce search space on top of the basic tree search with alpha-beta pruning for solving ultra-weak solutions. In particular, these consistencies approximate the lower and upper bounds of the cost of a problem by exploiting the semantics of the quantifiers and reusing techniques from both Weighted and Quantified CSPs. Lower bound computation employs standard estimation of costs in the sub-problems used in alpha-beta search. In estimating upper bounds, we propose two approaches based on the Duality Principle: duality of quantifiers and duality of constraints. The first duality amounts to changing quantifiers from min to max , while the second duality re-uses the lower bound approximation functions on dual constraints to generate upper bounds. Experiments on three benchmarks comparing basic alpha-beta pruning and the six consistencies from the two dualities are performed to confirm the feasibility and efficiency of our proposal.

Keywords

constraint optimization soft constraint satisfaction minimax game search consistency algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allis, L.V.: Searching for solutions in games and artificial intelligence. Ph.D. thesis, University of Limburg (1994)Google Scholar
  2. 2.
    Apt, K.: Principles of Constraint Programming. Cambridge University Press, New York (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benedetti, M., Lallouet, A., Vautard, J.: Quantified Constraint Optimization. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 463–477. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Bordeaux, L., Cadoli, M., Mancini, T.: CSP properties for quantified constraints: Definitions and complexity. In: AAAI 2005, pp. 360–365 (2005)Google Scholar
  5. 5.
    Bordeaux, L., Monfroy, E.: Beyond NP: Arc-Consistency for Quantified Constraints. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 371–386. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Brown, K.N., Little, J., Creed, P.J., Freuder, E.C.: Adversarial constraint satisfaction by game-tree search. In: ECAI 2004, pp. 151–155 (2004)Google Scholar
  7. 7.
    Cabon, B., de Givry, S., Lobjois, L., Schiex, T., Warners, J.: Radio link frequency assignment. Constraints 4, 79–89 (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cooper, M.C., de Givry, S., Schiex, T.: Optimal soft arc consistency. In: IJCAI 2007, pp. 68–73 (2007)Google Scholar
  9. 9.
    Cooper, M.C., de Givry, S., Sanchez, M., Schiex, T., Zytnicki, M., Werner, T.: Soft arc consistency revisited. Artificial Intelligence 174(7-8), 449–478 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Debruyne, R., Bessière, C.: From Restricted Path Consistency to Max-Restricted Path Consistency. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 312–326. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  11. 11.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers (2002)Google Scholar
  12. 12.
    Gent, I.P., Nightingale, P., Stergiou, K.: QCSP-Solve: A solver for quantified constraint satisfaction problems. In: IJCAI 2005, pp. 138–143 (2005)Google Scholar
  13. 13.
    van den Herik, H.J., Uiterwijk, J.W.H.M., van Rijswijck, J.: Games solved: Now and in the future. Artif. Intell. 134(1-2), 277–311 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Larrosa, J., Schiex, T.: In the quest of the best form of local consistency for weighted CSP. In: IJCAI 2003, pp. 239–244 (2003)Google Scholar
  15. 15.
    Larrosa, J., Schiex, T.: Solving weighted CSP by maintaining arc consistency. Artificial Intelligence 159(1-2), 1–26 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lee, J.H.M., Mak, T.W.K., Yip, J.: Weighted constraint satisfaction problems with min-max quantifiers. In: ICTAI 2011, pp. 769–776 (2011)Google Scholar
  17. 17.
    Lee, J.H.M., Shum, Y.W.: Modeling soft global constraints as linear programs in weighted constraint satisfaction. In: ICTAI 2011, pp. 305–312 (2011)Google Scholar
  18. 18.
    Lee, J.H.M., Leung, K.L.: Consistency techniques for flow-based projection-safe global cost functions in weighted constraint satisfaction. JAIR 43, 257–292 (2012)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Lee, J.H.M., Leung, K.L., Wu, Y.: Polynomially decomposable global cost functions in weighted constraint satisfaction. In: AAAI 2012 (to appear, 2012)Google Scholar
  20. 20.
    Mamoulis, N., Stergiou, K.: Algorithms for Quantified Constraint Satisfaction Problems. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 752–756. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Murty, K.G.: Linear and Combinatorial Programming. R. E. Krieger (1985)Google Scholar
  22. 22.
    Neumann, J.V., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press (1944)Google Scholar
  23. 23.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press (2007)Google Scholar
  24. 24.
    Russell, S.J., Norvig, P.: Artificial Intelligence: A Modern Approach. Pearson Education (2003)Google Scholar
  25. 25.
    Schaeffer, J., Burch, N., Björnsson, Y., Kishimoto, A., Müller, M., Lake, R., Lu, P., Sutphen, S.: Checkers is solved. Science 317(5844), 1518–1522 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Walsh, T.: Stochastic constraint programming. In: ECAI 2002, pp. 111–115 (2002)Google Scholar
  27. 27.
    Wolsey, L.A.: Integer Programming. Wiley (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Arnaud Lallouet
    • 1
  • Jimmy H. M. Lee
    • 2
  • Terrence W. K. Mak
    • 2
  1. 1.Université de Caen, GREYCCaen CedexFrance
  2. 2.The Chinese University of Hong KongShatinHong Kong

Personalised recommendations