From Pebble Games to Tractability: An Ambidextrous Consistency Algorithm for Quantified Constraint Satisfaction

  • Hubie Chen
  • Víctor Dalmau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)


The constraint satisfaction problem (CSP) and quantified constraint satisfaction problem (QCSP) are common frameworks for the modelling of computational problems. Although they are intractable in general, a rich line of research has identified restricted cases of these problems that are tractable in polynomial time. Remarkably, many tractable cases of the CSP that have been identified are solvable by a single algorithm, which we call here the consistency algorithm. In this paper, we give a natural extension of the consistency algorithm to the QCSP setting, by making use of connections between the consistency algorithm and certain two-person pebble games. Surprisingly, we demonstrate a variety of tractability results using the algorithm, revealing unified structure among apparently different cases of the QCSP.


Relational Structure Constraint Satisfaction Constraint Satisfaction Problem Winning Strategy Relation Symbol 
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  1. 1.
    Bulatov, A.: Combinatorial problems raised from 2-semilattices (Manuscript)Google Scholar
  2. 2.
    Bulatov, A.: A dichotomy theorem for constraints on a three-element set. In: Proceedings of 43rd IEEE Symposium on Foundations of Computer Science, pp. 649–658 (2002)Google Scholar
  3. 3.
    Bulatov, A.: Tractable conservative constraint satisfaction problems. In: Proceedings of 18th IEEE Symposium on Logic in Computer Science (LICS 2003), pp. 321–330 (2003); Extended version appears as Oxford University technical report PRG-RR–03-01Google Scholar
  4. 4.
    Bulatov, A.: A graph of a relational structure and constraint satisfaction problems. In: Proceedings of 19th IEEE Annual Symposium on Logic in Computer Science, LICS 2004 (2004)Google Scholar
  5. 5.
    Chandra, A., Merlin, P.: Optimal implementation of conjunctive queries in relational data bases. In: STOC (1977)Google Scholar
  6. 6.
    Chen, H.: The Computational Complexity of Quantified Constraint Satisfaction. PhD thesis, Cornell University (August 2004)Google Scholar
  7. 7.
    Chen, H.: Quantified constraint satisfaction and bounded treewidth. In: ECAI (2004)Google Scholar
  8. 8.
    Chen, H.: Quantified constraint satisfaction, maximal constraint languages, and symmetric polymorphisms. In: STACS (2005)Google Scholar
  9. 9.
    Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, p. 310. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Dalmau, V., Pearson, J.: Closure functions and width 1 problems. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 159–173. Springer, Heidelberg (1999)Google Scholar
  11. 11.
    Dechter, R., Pearl, J.: Tree clustering for constraint networks. Artificial Intelligence, pp. 353–366 (1989)Google Scholar
  12. 12.
    Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. JACM (2002)Google Scholar
  13. 13.
    Freuder, E.: Complexity of k-tree structured constraint satisfaction problems. In: AAAI 1990 (1990)Google Scholar
  14. 14.
    Gottlob, G., Greco, G., Scarcello, F.: The complexity of quantified constraint satisfaction problems under structural restrictions. In: IJCAI (2005)Google Scholar
  15. 15.
    Gottlob, G., Leone, N., Scarcello, F.: A comparison of structural csp decomposition methods. Artif. Intell. 124(2), 243–282 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. In: FOCS 2003, pp. 552–561 (2003)Google Scholar
  17. 17.
    Jeavons, P., Cohen, D., Cooper, M.: Constraints, consistency, and closure. Articial Intelligence 101(1-2), 251–265 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kolaitis, P.G., Vardi, M.Y.: On the expressive power of Datalog: tools and a case study. Journal of Computer and System Sciences 51(1), 110–134 (1995)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. Journal of Computer and System Sciences 61, 302–332 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kolaitis, P.G., Vardi, M.Y.: A game-theoretic approach to constraint satisfaction. In: Proceedings 17th National (US) Conference on Artificial Intellignece, AAAI 2000, pp. 175–181 (2000)Google Scholar
  21. 21.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 216–226 (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hubie Chen
    • 1
  • Víctor Dalmau
    • 1
  1. 1.Departament de TecnologiaUniversitat Pompeu FabraBarcelonaSpain

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