Theory of Computing Systems

, Volume 48, Issue 3, pp 444–464 | Cite as

Tractable Structures for Constraint Satisfaction with Truth Tables

  • Dániel Marx


The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure adaptive width and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case. Finally, we present a class of hypergraphs with bounded adaptive width and unbounded fractional hypertree width.

Computational complexity Constraint satisfaction Treewidth Adaptive width 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, I.: Width functions for hypertree decompositions. PhD thesis, Albert-Ludwigs-Universität Freiburg (2006) Google Scholar
  2. 2.
    Bulatov, A.A.: Tractable conservative constraint satisfaction problems. In: 18th Annual IEEE Symposium on Logic in Computer Science (LICS’03), p. 321. IEEE Computer Society, Los Alamitos (2003) Google Scholar
  3. 3.
    Bulatov, A.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.A., Krokhin, A.A., Jeavons, P.: The complexity of maximal constraint languages. In: Proceedings of the 33rd ACM Symposium on Theory of Computing, pp. 667–674 (2001) Google Scholar
  5. 5.
    Chen, H., Grohe, M.: Constraint satisfaction problems with succinctly specified relations. (2006). Manuscript. Preliminary version in Dagstuhl Seminar Proceedings 06401: Complexity of Constraints Google Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999) Google Scholar
  7. 7.
    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 (1999) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  9. 9.
    Freuder, E.C.: Complexity of k-tree structured constraint satisfaction problems. In: Proc. of AAAI-90, pp. 4–9, Boston, MA (1990) Google Scholar
  10. 10.
    Gottlob, G., Leone, N., Scarcello, F.: Hypertree decompositions and tractable queries. J. Comput. Syst. Sci. 64, 579–627 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gottlob, G., Scarcello, F., Sideri, M.: Fixed-parameter complexity in AI and nonmonotonic reasoning. Artif. Intell. 138(1–2), 55–86 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Grohe, M.: The structure of tractable constraint satisfaction problems. In: MFCS 2006, pp. 58–72 (2006) Google Scholar
  13. 13.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1), 1 (2007) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Grohe, M., Marx, D.: Constraint solving via fractional edge covers. In: SODA’06: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 289–298. ACM, New York (2006) CrossRefGoogle Scholar
  15. 15.
    Grohe, M., Schwentick, T., Segoufin, L.: When is the evaluation of conjunctive queries tractable? In: STOC’01: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 657–666. ACM, New York (2001) CrossRefGoogle Scholar
  16. 16.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jeavons, P., Cohen, D.A., Gyssens, M.: Closure properties of constraints. J. ACM 44(4), 527–548 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci. 61(2), 302–332 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Marx, D.: Can you beat treewidth? In: 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07), pp. 169–179 (2007) Google Scholar
  20. 20.
    Marx, D.: Approximating fractional hypertree width. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’09) (2009) Google Scholar
  21. 21.
    Scarcello, F., Gottlob, G., Greco, G.: Uniform constraint satisfaction problems and database theory. In: Complexity of Constraints, pp. 156–195 (2008) Google Scholar
  22. 22.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Conference Record of the Tenth Annual ACM Symposium on Theory of Computing, San Diego, CA, 1978, pp. 216–226. ACM, New York (1978) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Tel Aviv UniversityTel AvivIsrael

Personalised recommendations