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International Conference on Principles and Practice of Constraint Programming

CP 2003: Principles and Practice of Constraint Programming – CP 2003 pp 392–406Cite as

Tractability by Approximating Constraint Languages

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2833))

Abstract

A constraint satisfaction problem instance consists of a collection of variables that need to have values assigned to them. The assignments are limited by constraints that force the values taken by certain collections of variables (the constraint scopes) to satisfy specified properties (the constraint relations).

As the general CSP problem is NP-hard there has been significant effort devoted to discovering tractable subproblems of the CSP.

The structure of a CSP instance is defined to be the hypergraph formed by the constraint scopes. Restricting the possible structure of the CSP instances has been a successful way of identifying tractable subproblems.

The language of a CSP instance is defined to be the set of constraint relations of the instance. Restricting the language allowed for CSP instances has also yielded many interesting tractable subproblems.

Almost all known tractable subproblems are either structural or relational. In this paper we construct tractable subproblems of the general CSP that are neither defined by structural nor relational properties.

These new tractable classes are related to tractable languages in much the same way that general decompositions (cutset, tree-clustering, etc.) are related to acyclic decompositions. It may well be that our results will begin to make language based tractability of more practical interest.

We show that our theory allows us to properly extend the binary max-closed language based tractable class, which is maximal as a tractable binary constraint language. Our theory also explains the tractability of the constraint representation of the Stable Marriage Problem which has not been amenable to existing explanations of tractability. In fact we provide a uniform explanation for the tractability of the class of max-closed CSPs and the SMP.

There has been much work done on so called renamable HORN theories which are a tractable subproblem of SAT. It has been shown that renamable HORN theories are tractably identifiable and solvable. It has also been shown that finding the largest sub-theory that is renamable HORN is NP-hard. These results also follow immediately from our theory.

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References

  1. Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear time algorithm for testing the truth of certain quantified Boolean formulas. Information Processing Letters 8, 121–123 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aspvall, B.: Recognizing disguised NR(1) instances of the satisfiability problem. Journal of Algorithms 1(1), 97–103 (1980) (note)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. Journal of the ACM 30, 479–513 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bessière, C., Régin, J.-C.: Arc consistency for general constraint networks: preliminary results. In: Proceedings of IJCAI 1997, Nagoya, Japan, pp. 398–404 (1997)

    Google Scholar 

  5. Boros, E.: Maximum renamable Horn sub-CNFs. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science  96 (1999)

    Google Scholar 

  6. Chandru, V., Coullard, C., Hammer, P., Monta nez, M., Sun, X.: On renamable Horn and generalized Horn functions. Annals of Mathematics and Artificial Intelligence 1, 33–48 (1990)

    Article  MATH  Google Scholar 

  7. Chandru, V., Hooker, J.N.: Extended Horn sets in propositional logic. Journal of the ACM 38, 205–221 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. Journal of Computer and System Sciences 51, 511–522 (1995)

    Article  MathSciNet  Google Scholar 

  9. Dechter, R., Pearl, J.: Network-based heuristics for constraint satisfaction problems. Artificial Intelligence 34(1), 1–38 (1988)

    Article  MathSciNet  Google Scholar 

  10. Dechter, R., Pearl, J.: Tree clustering for constraint networks. Artificial Intelligence 38, 353–366 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. del Val, A.: On 2-SAT and renamable Horn. In: AAAI: 17th National Conference on Artificial Intelligence. AAAI / MIT Press (2000)

    Google Scholar 

  12. Fagin, R.: Degrees of acyclicity for hypergraphs and relational database schemes. Journal of the ACM 30, 514–550 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gale, D., Shapley, L.S.: College admissions and stability of marriage. Amer. Math. Monthly 69(1), 9–15 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gault, R.L.: Polyanna technical manual (version 1.00). Technical Report RR-01-20, Computing Laboratory. Oxford University (2001)

    Google Scholar 

  15. Gent, I.P., Irving, R.W., Manlove, M.F., Prosser, P., Smith, B.M.: A constraint programming approach to the stable marriage problem. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 225–239. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  16. Gottlob, G., Leone, L., Scarcello, F.: Hypertree decompositions: A survey. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 37–57. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  17. Gottlob, G., Leone, L., Scarcello, F.: Hypertree decomposition and tractable queries. Journal of Computer and System Sciences 64(3), 579–627 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Green, M.J., Cohen, D.A.: Tractability by approximating constraint languages. Technical Report CSD-TR-03-01, Department of Computer Science, Royal Holloway, University of London, Egham, Surrey, UK (2003)

    Google Scholar 

  19. Gyssens, M., Jeavons, P.G., Cohen, D.A.: Decomposing constraint satisfaction problems using database techniques. Artificial Intelligence 66(1), 57–89 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jeavons, P.G., Cohen, D.A., Gyssens, M.: Closure properties of constraints. Journal of the ACM 44, 527–548 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jeavons, P.G., Cooper, M.C.: Tractable constraints on ordered domains. Artificial Intelligence 79(2), 327–339 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lewis, H.R.: Renaming a set of clauses as a Horn set. JACM 25(1), 134–135 (1978)

    Article  MATH  Google Scholar 

  23. Mackworth, A.K.: Consistency in networks of relations. Artificial Intelligence 8, 99–118 (1977)

    Article  MATH  Google Scholar 

  24. Montanari, U.: Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences 7, 95–132 (1974)

    Article  MathSciNet  Google Scholar 

  25. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings 10th ACM Symposium on Theory of Computing, STOC 1978, pp. 216–226 (1978)

    Google Scholar 

  26. van Beek, P., Dechter, R.: On the minimality and decomposability of row-convex constraint networks. Journal of the ACM 42, 543–561 (1995)

    Article  MATH  Google Scholar 

  27. van Dongen, M.R.C.: From local closure properties to a global closure property. Personal communication (2002; The source of the SMP example

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Royal Holloway, University of London, UK

    Martin J. Green & David A. Cohen

Authors
  1. Martin J. Green
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  2. David A. Cohen
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121, Padova, Italy

    Francesca Rossi

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Green, M.J., Cohen, D.A. (2003). Tractability by Approximating Constraint Languages. In: Rossi, F. (eds) Principles and Practice of Constraint Programming – CP 2003. CP 2003. Lecture Notes in Computer Science, vol 2833. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45193-8_27

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  • DOI: https://doi.org/10.1007/978-3-540-45193-8_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20202-8

  • Online ISBN: 978-3-540-45193-8

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