Subexponential Time Complexity of CSP with Global Constraints

  • Ronald de Haan
  • Iyad Kanj
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8656)

Abstract

Not all NP-complete problems share the same practical hardness with respect to exact computation. Whereas some NP-complete problems are amenable to efficient computational methods, others are yet to show any such sign. It becomes a major challenge to develop a theoretical framework that is more fine-grained than the theory of NP-completeness, and that can explain the distinction between the exact complexities of various NP-complete problems. This distinction is highly relevant for constraint satisfaction problems under natural restrictions, where various shades of hardness can be observed in practice.

Acknowledging the NP-hardness of such problems, one has to look beyond polynomial time computation. The theory of subexponential time complexity provides such a framework, and has been enjoying increasing popularity in complexity theory. Recently, a first analysis of the subexponential time complexity of classical CSPs (where all constraints are given extensionally as tables) was given.

In this paper, we extend this analysis to CSPs in which constraints are given intensionally in the form of global constraints. In particular, we consider CSPs that use the fundamental global constraints AllDifferent, AtLeastNValue, AtMost- NValue, and constraints that are specified by compressed tuples (cTable). We provide tight characterizations of the subexponential time complexity of the aforementioned problems with respect to several natural structural parameters.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beldiceanu, N., Carlsson, M., Rampon, J.-X.: Global constraint catalog. Technical Report T2005:08, SICS, SE-16 429 Kista, Sweden (August 2006), http://www.emn.fr/x-info/sdemasse/gccat/
  2. 2.
    Bessiere, C., Hebrard, E., Hnich, B., Kiziltan, Z., Walsh, T.: Filtering algorithms for the NValue constraint. Constraints 11(4), 271–293 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bessière, C., Hebrard, E., Hnich, B., Walsh, T.: The complexity of global constraints. In: McGuinness, D.L., Ferguson, G. (eds.) Proceedings of the Nineteenth National Conference on Artificial Intelligence, San Jose, California, USA, July 25-29, pp. 112–117. AAAI Press / The MIT Press (2004)Google Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Brooks, R.L.: On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society 37, 194–197 (1941)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, H., Grohe, M.: Constraint satisfaction with succinctly specified relations. J. of Computer and System Sciences 76(8), 847–860 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong computational lower bounds via parameterized complexity. J. of Computer and System Sciences 72(8), 1346–1367 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Chen, J., Kanj, I., Perkovic, L., Sedgwick, E., Xia, G.: Genus characterizes the complexity of certain graph problems: Some tight results. Journal of Computer and System Sciences 73(6), 892–907 (2007)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chen, J., Kanj, I.A., Xia, G.: On parameterized exponential time complexity. Theoretical Computer Science 410(27-29), 2641–2648 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)MATHGoogle Scholar
  11. 11.
    Dechter, R.: Constraint Processing. Morgan Kaufmann (2003)Google Scholar
  12. 12.
    Demaine, E., Fomin, F., Hajiaghayi, M., Thilikos, D.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM 52, 866–893 (2005)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Fellows, M.R., Friedrich, T., Hermelin, D., Narodytska, N., Rosamond, F.A.: Constraint satisfaction problems: Convexity makes alldifferent constraints tractable. In: Walsh, T. (ed.) IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, pp. 522–527. IJCAI/AAAI (2011)Google Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)Google Scholar
  15. 15.
    Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. of the ACM 29(1), 24–32 (1982)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Freuder, E.C.: Complexity of k-tree structured constraint satisfaction problems. In: Shrobe, H.E., Dietterich, T.G., Swartout, W.R. (eds.) Proceedings of the 8th National Conference on Artificial Intelligence, Boston, Massachusetts, July 29-August 3, 2 vols., pp. 4–9. AAAI Press / The MIT Press (1990)Google Scholar
  17. 17.
    Garey, M.R., Johnson, D.R.: Computers and Intractability. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
  18. 18.
    Gaspers, S., Szeider, S.: Kernels for global constraints. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, pp. 540–545. AAAI Press/IJCAI (2011)Google Scholar
  19. 19.
    Gottlob, G., Leone, N., Scarcello, F.: Hypertree decompositions and tractable queries. J. of Computer and System Sciences 64(3), 579–627 (2002)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Hnich, B., Kiziltan, Z., Walsh, T.: Combining symmetry breaking with other constraints: Lexicographic ordering with sums. In: AI&M 1-2004, Eighth International Symposium on Artificial Intelligence and Mathematics, Fort Lauderdale, Florida, USA, January 4-6 (2004)Google Scholar
  21. 21.
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. of Computer and System Sciences 62(2), 367–375 (2001)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. of Computer and System Sciences 63(4), 512–530 (2001)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Kanj, I., Szeider, S.: On the subexponential time complexity of CSP. In: Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence. AAAI Press (2013)Google Scholar
  24. 24.
    Katsirelos, G., Walsh, T.: A compression algorithm for large arity extensional constraints. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 379–393. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  25. 25.
    Kolaitis, P.G., Vardi, M.Y.: Conjunctive-query containment and constraint satisfaction. J. of Computer and System Sciences 61(2), 302–332 (2000); Special issue on the Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, Seattle, WA (1998)Google Scholar
  26. 26.
    Kutz, M., Elbassioni, K., Katriel, I., Mahajan, M.: Simultaneous matchings: hardness and approximation. J. of Computer and System Sciences 74(5), 884–897 (2008)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bulletin of the European Association for Theoretical Computer Science 105, 41–72 (2011)MATHMathSciNetGoogle Scholar
  28. 28.
    Marx, D.: Tractable hypergraph properties for constraint satisfaction and conjunctive queries. J. of the ACM 60(6), Art. 42, 51 (2013)Google Scholar
  29. 29.
    Pachet, F., Roy, P.: Automatic generation of music programs. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 331–345. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  30. 30.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. of Computer and System Sciences 43(3), 425–440 (1991)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Régin, J.-C.: A filtering algorithm for constraints of difference in CSPs. In: Hayes-Roth, B., Korf, R.E. (eds.) Proceedings of the 12th National Conference on Artificial Intelligence, Seattle, WA, USA, July 31-August 4, vol. 1, pp. 362–367. AAAI Press / The MIT Press (1994)Google Scholar
  32. 32.
    Régin, J.-C.: Développement d’outils algorithmiques pour l’Intelligence Artificielle. PhD thesis, Montpellier II (1995) (in French)Google Scholar
  33. 33.
    Régin, J.-C.: Global constraints: A survey. In: van Hentenryck, P., Milano, M. (eds.) Hybrid Optimization: The Ten Years of CPAIOR. Optimization and Its Applications, vol. 45, ch. 3, pp. 63–134. Springer (2011)Google Scholar
  34. 34.
    Régin, J.-C., Rueher, M.: A global constraint combining a sum constraint and difference constraints. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 384–395. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  35. 35.
    Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. of Computer and System Sciences 76(2), 103–114 (2010)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    van Hoeve, W.-J., Katriel, I.: Global constraints. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, ch. 6. Elsevier (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ronald de Haan
    • 1
  • Iyad Kanj
    • 2
  • Stefan Szeider
    • 1
  1. 1.Institute of Information SystemsVienna University of TechnologyViennaAustria
  2. 2.School of ComputingDePaul UniversityChicagoUSA

Personalised recommendations