Advertisement

Constraint Satisfaction with Counting Quantifiers 2

  • Barnaby Martin
  • Juraj Stacho
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8476)

Abstract

We study constraint satisfaction problems (CSPs) in the presence of counting quantifiers ∃  ≥ j , asserting the existence of j distinct witnesses for the variable in question. As a continuation of our previous (CSR 2012) paper [11], we focus on the complexity of undirected graph templates. As our main contribution, we settle the two principal open questions proposed in [11]. Firstly, we complete the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, placing each case either in P, NP-complete or Pspace-complete. This involves resolution of the cases in which we have the single quantifier ∃  ≥ j on the clique \(\mathbb{K}_{2j}\). Secondly, we confirm a conjecture from [11], which proposes a full dichotomy for ∃ and ∃  ≥ 2 on all finite undirected graphs. The main thrust of this second result is the solution of the complexity for the infinite path which we prove is a polynomial-time solvable problem. By adapting the algorithm for the infinite path we are then able to solve the problem for finite paths, and then trees and forests. Thus as a corollary to this work, combining with the other cases from [11], we obtain a full dichotomy for ∃ and ∃  ≥ 2 quantifiers on finite graphs, each such problem being either in P or NP-hard. Finally, we persevere with the work of [11] in exploring cases in which there is dichotomy between P and Pspace-complete, and contrast this with situations in which the intermediate NP-completeness may appear.

Keywords

Bipartite Graph Undirected Graph Constraint Satisfaction Total Order Constraint Satisfaction Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell). SIAM Journal on Computing 38(5), 1782–1802 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Börner, F., Bulatov, A.A., Chen, H., Jeavons, P., Krokhin, A.A.: The complexity of constraint satisfaction games and QCSP. Inf. Comput. 207(9), 923–944 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bulatov, A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53(1), 66–120 (2006)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bulatov, A., Krokhin, A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34, 720–742 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bulatov, A.A., Hedayaty, A.: Counting predicates, subset surjective functions, and counting csps. In: 42nd IEEE International Symposium on Multiple-Valued Logic, ISMVL 2012, pp. 331–336 (2012)Google Scholar
  6. 6.
    Bulatov, A.A., Hedayaty, A.: Galois correspondence for counting quantifiers. CoRR abs/1210.3344 (2012)Google Scholar
  7. 7.
    Feder, T., Vardi, M.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing 28, 57–104 (1999)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kolaitis, P.G., Vardi, M.Y.: A logical Approach to Constraint Satisfaction. In: Finite Model Theory and Its Applications. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag New York, Inc. (2005)Google Scholar
  10. 10.
    Madelaine, F., Martin, B.: QCSP on Partially Reflexive Cycles – The Wavy Line of Tractability. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013. LNCS, vol. 7913, pp. 322–333. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Madelaine, F., Martin, B., Stacho, J.: Constraint Satisfaction with Counting Quantifiers. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 253–265. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Martin, B.: QCSP on partially reflexive forests. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 546–560. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of STOC 1978, pp. 216–226 (1978)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Barnaby Martin
    • 1
  • Juraj Stacho
    • 2
  1. 1.School of Science and TechnologyMiddlesex UniversityLondonU.K.
  2. 2.IEOR DepartmentColumbia UniversityNew YorkUnited States

Personalised recommendations