A Dichotomy Theorem for Typed Constraint Satisfaction Problems

  • Su Chen
  • Tomasz Imielinski
  • Karin Johnsgard
  • Donald Smith
  • Mario Szegedy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4121)


This paper is a contribution to the general investigation into how the complexity of constraint satisfaction problems (CSPs) is determined by the form of the constraints. Schaefer proved that the Boolean generalized CSP has the dichotomy property (i.e., all instances are either in P or are NP-complete), and gave a complete and simple classification of those instances which are in P (assuming \(\mbox{P}\neq\mbox{NP}\)) [20]. In this paper we consider a special subcase of the generalized CSP. For this CSP subcase, we require that the variables be drawn from disjoint Boolean domains. Our relation set contains only two elements: a monotone multiple-arity Boolean relation R and its complement \(\overline{R}\). We prove a dichotomy theorem for these monotone function CSPs, and characterize those monotone functions such that the corresponding problem resides in P.


Boolean Function Monotone Function Constraint Satisfaction Problem Dichotomy Theorem Truth Assignment 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berge, C.: Hypergraphs: Combinatorics of Finite Sets. North-Holland, Amsterdam (1989)zbMATHGoogle Scholar
  2. 2.
    Bulatov, A.A.: Tractable conservative constraint satisfaction problems. ACM Transactions on Computational Logic (submitted)Google Scholar
  3. 3.
    Bulatov, A.A.: A dichotomy theorem for constraints on a three-element set. In: Proceedings of 43rd Annual IEEE Symposium of Foundation of Computer Science (FOCS 2002), pp. 649–658 (2002)Google Scholar
  4. 4.
    Bulatov, A.A.: Tractable conservative constraint satisfaction problems. In: Proceedings of 18th IEEE Symposium on Logic in Computer Science (LICS 2003), pp. 321–330 (2003)Google Scholar
  5. 5.
    Bulatov, A.A., Jeavons, P.: An algebraic approach to multi-sorted constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 183–198. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Bulatov, A.A., Krokhin, A.A., Jeavons, P.: Constraint satisfaction problems and finite algebras. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 272–282. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Cohen, D., Jeavons, P.: Handbook of Constraint Programming. In: The complexity of constraint languages, ch.6 Elsevier (to appear, 2006)Google Scholar
  8. 8.
    Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. J. Comput. Syst. Sci. 51(3), 511–522 (1995)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001)zbMATHCrossRefGoogle Scholar
  10. 10.
    Dalmau, V., Ford, D.K.: Generalized satisfiability with limited occurrences per variable: A study through delta-matroid parity. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 358–367. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Feder, T.: Fanout limitations on constraint systems. Theor. Comput. Sci. 255(1-2), 281–293 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Feder, T., Ford, D.: Classification of bipartite boolean constraint satisfaction through delta-matroid intersection. Electronic Colloquium on Computational Complexity (ECCC), TR05(016) (2005) (to appear in SIAM J. Discrete Math)Google Scholar
  13. 13.
    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
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co., New York (1979)zbMATHGoogle Scholar
  15. 15.
    Grohe, M.: The complexity of homomorphism and onstraint satisfaction problems seen from the other side. In: 44th Symposium on Foundations of Computer Science (FOCS 2003), pp. 552–561 (2003)Google Scholar
  16. 16.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Comb. Theory, Series B 48, 92–110 (1990)zbMATHCrossRefGoogle Scholar
  17. 17.
    Istrate, G.: Looking for a version of Schaefer’s dichotomy theorem when each variable occurs at most twice. Technical Report TR 652, U. Rochester, CS Dept. (1997)Google Scholar
  18. 18.
    Jeavons, P.G.: On the algebraic structure of combinatorial problems. Theor. Comput. Sci. 200(1–2), 185–204 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ladner, R.E.: On the structure of polynomial time reducibility. J. ACM 22(1), 155–171 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226 (1978)Google Scholar
  21. 21.
    Wegener, I.: Complexity of Boolean Functions. John Wiley and Sons, Chichester (1987)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Su Chen
    • 1
  • Tomasz Imielinski
    • 1
  • Karin Johnsgard
    • 2
  • Donald Smith
    • 1
  • Mario Szegedy
    • 1
  1. 1.Rutgers UniversityPiscatawayUSA
  2. 2.Monmouth UniversityWest Long BranchUSA

Personalised recommendations