On the Complexity of Existential Pebble Games

  • Phokion G. Kolaitis
  • Jonathan Panttaja
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2803)


Existential k-pebble games, k≥ 2, are combinatorial games played between two players, called the Spoiler and the Duplicator, on two structures. These games were originally introduced in order to analyze the expressive power of Datalog and related infinitary logics with finitely many variables. More recently, however, it was realized that existential k-pebble games have tight connections with certain consistency properties which play an important role in identifying tractable classes of constraint satisfaction problems and in designing heuristic algorithms for solving such problems. Specifically, it has been shown that strong k-consistency can be established for an instance of constraint satisfaction if and only if the Duplicator has a winnning strategy for the existential k-pebble game between two finite structures associated with the given instance of constraint satisfaction. In this paper, we pinpoint the computational complexity of determining the winner of the existential k-pebble game. The main result is that the following decision problem is EXPTIME-complete: given a positive integer k and two finite structures A and B, does the Duplicator win the existential k-pebble game on A and B? Thus, all algorithms for determining whether strong k-consistency can be established (when k is part of the input) are inherently exponential.


Undirected Graph Constraint Satisfaction Constraint Satisfaction Problem Expressive Power Winning Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the ACM 28, 114–233 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cooper, M.C.: An optimal k-consistency algorithm. Artificial Intelligence 41, 89–95 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Proc. of Eighth International Conference on Principles and Practice of Constraint Programming, pp. 310–326 (2002)Google Scholar
  4. 4.
    Dechter, R.: Constraint networks. In: Shapiro, S.C. (ed.) Encyclopedia of Artificial Intelligence, pp. 276–285. Wiley, NewYork (1992)Google Scholar
  5. 5.
    Dechter, R.: From local to global consistency. Artificial Intelligence 55, 87–107 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM Journal on Computing 28, 57–104 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Grohe, M.: Equivalence in finite-variable logics is complete for polynomial time. Combinatorica 19(4), 507–523 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kasai, T., Adachi, A., Iwata, S.: Classes of pebble games and complete problems. SIAM Journal of Computing 8(4), 574–586 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kasai, T., Iwata, S.: Gradually intractable problems and nondeterminitstic log-space lower bounds. Mathematical Systems Theory 18, 153–170 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kasif, S.: On the parallel complexity of some constraint satisfaction problems. In: Proc. of Fifth National Conference on Artificial Intelligencee, vol. 1, pp. 349–353 (1986)Google Scholar
  11. 11.
    Kolaitis, P.G., Vardi, M.Y.: On the expressive power of Datalog: Tools and a case study. Journal of Computer and System Sciences 51, 110–134 (1995)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kolaitis, P.G., Vardi, M.Y.: A game-theoretic approach to constraint satisfaction. In: Proc. of the Seventeenth National Conference on Artificial Intelligence, pp. 175–181 (2000)Google Scholar
  13. 13.
    Pezzoli, E.: Computational complexity of Ehrenfeucht-Fraïssé games on finite structures. In: Gottlob, G., Grandjean, E., Seyr, K. (eds.) CSL 1998. LNCS, vol. 1584, pp. 159–170. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Phokion G. Kolaitis
    • 1
  • Jonathan Panttaja
    • 1
  1. 1.Computer Science DepartmentUniversity of CaliforniaSanta Cruz

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