Quantified Valued Constraint Satisfaction Problem

  • Florent MadelaineEmail author
  • Stéphane Secouard
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11008)


We study the complexity of the quantified and valued extension of the constraint satisfaction problem (QVCSP) for certain classes of languages. This problem is also known as the weighted constraint satisfaction problem with min-max quantifiers [1].

The multimorphisms that preserve a language is the starting point of our analysis. We establish some situations where a QVCSP is solvable in polynomial time by formulating new algorithms or by extending the usage of collapsibility, a property well known for reducing the complexity of the quantified CSP (QCSP) from Pspace to NP. In contrast, we identify some classes of problems for which the VCSP is tractable but the QVCSP is Pspace-hard.

As a main Corollary, we derive an analogue of Shaeffer’s dichotomy between P and Pspace for QCSP on Boolean languages and Cohen et al. dichotomy between P and NP-complete for VCSP on Boolean valued languages: we prove that the QVCSP follows a dichotomy between P and Pspace-complete.

Finally, we exhibit examples of NP-complete QVCSP for domains of size 3 and more, which suggest at best a trichotomy between P, NP-complete and Pspace-complete for the QVCSP.


Complexity classification Valued CSP Quantified CSP Polymorphisms Multimorphisms Collapsibility 



The authors are thankful to the three anonymous reviewers for their valuable comments which have helped us improve the manuscript.


  1. 1.
    Lee, J.H., Mak, T.W.K., Yip, J.: Weighted constraint satisfaction problems with min-max quantifiers. In: IEEE 23rd International Conference on Tools with Artificial Intelligence, ICTAI 2011, Boca Raton, FL, USA, 7–9 November 2011, pp. 769–776. IEEE Computer Society (2011).
  2. 2.
    Beyond np. Accessed 21 June 2017
  3. 3.
    Bulatov, A.A.: A dichotomy theorem for nonuniform CSPs. In: Umans [26], pp. 319–330.
  4. 4.
    Zhuk, D.: A proof of CSP dichotomy conjecture. In: Umans [26], pp. 331–342.
  5. 5.
    Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1998). Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Martin, B.: QCSP on partially reflexive forests. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 546–560. Springer, Heidelberg (2011). Scholar
  8. 8.
    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). Scholar
  9. 9.
    Dapic, P., Markovic, P., Martin, B.: Quantified constraint satisfaction problem on semicomplete digraphs. ACM Trans. Comput. Log. 18(1), 2:1–2:47 (2017). Scholar
  10. 10.
    Thapper, J., Zivny, S.: The complexity of finite-valued CSPs. J. ACM 63(4), 37:1–37:33 (2016). Scholar
  11. 11.
    Kolmogorov, V., Krokhin, A.A., Rolinek, M.: The complexity of general-valued CSPs. In: Guruswami, V. (ed.) IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17–20 October 2015, pp. 1246–1258. IEEE Computer Society (2015).
  12. 12.
    Martin, B.: Quantified constraints in twenty seventeen. In: Krokhin and Zivny [25], pp. 327–346.
  13. 13.
    Krokhin, A.A., Zivny, S.: The complexity of valued CSPs. In: The Constraint Satisfaction Problem: Complexity and Approximability [25], pp. 233–266.
  14. 14.
    Schaefer, T.: The complexity of satisfiability problems. In: STOC (1978)Google Scholar
  15. 15.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. Society for Industrial and Applied Mathematics, Philadelphia (2001)CrossRefGoogle Scholar
  16. 16.
    Dalmau, V.: Some dichotomy theorems on constant-free quantified boolean formulas. Technical report LSI-97-43-R., Departament LSI, Universitat Pompeu Fabra (1997)Google Scholar
  17. 17.
    Cohen, D.A., Cooper, M.C., Jeavons, P., Krokhin, A.A.: The complexity of soft constraint satisfaction. Artif. Intell. 170(11), 983–1016 (2006). Scholar
  18. 18.
    Benedetti, M., Lallouet, A., Vautard, J.: Modeling adversary scheduling with qcsp\({}^{\text{+}}\). In: Wainwright, R.L., Haddad, H. (eds.) Proceedings of the 2008 ACM Symposium on Applied Computing (SAC), Fortaleza, Ceara, Brazil, 16–20 March 2008, pp. 151–155. ACM (2008).
  19. 19.
    Chen, H.: The complexity of quantified constraint satisfaction: collapsibility, sink algebras, and the three-element case. SIAM J. Comput. 37(5), 1674–1701 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Chen, H.: Quantified constraint satisfaction and the polynomially generated powers property. Algebra Universalis 65(3), 213–241 (2011). an extended abstract appeared in ICALP B 2008MathSciNetCrossRefGoogle Scholar
  21. 21.
    Carvalho, C., Madelaine, F.R., Martin, B.: From complexity to algebra and back: digraph classes, collapsibility, and the PGP. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, 6–10 July 2015, pp. 462–474. IEEE Computer Society (2015).
  22. 22.
    Carvalho, C., Martin, B., Zhuk, D.: The complexity of quantified constraints. CoRR abs/1701.04086 (2017).
  23. 23.
    Chen, H.: Meditations on quantified constraint satisfaction. CoRR abs/1201.6306 (2012)CrossRefGoogle Scholar
  24. 24.
    Martin, B., Madelaine, F.: Towards a trichotomy for quantified H-Coloring. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 342–352. Springer, Heidelberg (2006). Scholar
  25. 25.
    Krokhin, A.A., Zivny, S. (eds.): The Constraint Satisfaction Problem: Complexity and Approximability, Dagstuhl Follow-Ups, vol. 7. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017).
  26. 26.
    Umans, C. (ed.): 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, 15–17 October 2017. IEEE Computer Society (2017).

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Authors and Affiliations

  1. 1.Université Paris-Est Créteil, LACLCréteilFrance
  2. 2.Université Caen Normandie, CNRS, GREYCCaenFrance

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