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Abstract

We study two natural models of randomly generated constraint satisfaction problems. We determine how quickly the domain size must grow with n to ensure that these models are robust in the sense that they exhibit a non-trivial threshold of satisfiability, and we determine the asymptotic order of that threshold. We also provide resolution complexity lower bounds for these models.

Keywords

Random Graph Small Tree Constraint Satisfaction Constraint Satisfaction Problem Random Model 
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.

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References

  1. 1.
    Achlioptas, D., Beame, P., Molloy, M.: A sharp threshold in proof complexity. In: Proceedings of STOC, pp. 337–346 (2001)Google Scholar
  2. 2.
    Achlioptas, D., Kirousis, L., Kranakis, E., Krizanc, D., Molloy, M., Stamatiou, Y.: Random constraint satisfaction: a more accurate picture. Constraints 6, 324–329 (2001); Conference version in Proceedings of CP 1997, 107–120Google Scholar
  3. 3.
    Beame, P., Culberson, J., Mitchell, D.: The resolution complexity of random graph k-colourability (in preparation) Google Scholar
  4. 4.
    Beame, P., Pitassi, T.: Simplified and improved resolution lower bounds. In: Proceedings of FOCS, pp. 274–282 (1996)Google Scholar
  5. 5.
    Beame, P., Karp, R., Pitassi, T., Saks, M.: The efficiency of resolution and Davis- Putnam procedures. In: Proceedings of STOC 1998 and SIAM Journal on Computing, vol. 31, pp. 1048–1075 (2002)Google Scholar
  6. 6.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. In: Proceedings of STOC 1999 and Journal of the ACM, vol. 48 (2001)Google Scholar
  7. 7.
    Bollobás, B.: Random graphs, 2nd edn. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  8. 8.
    Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal on Combinatorics 1, 311–316 (1980)zbMATHGoogle Scholar
  9. 9.
    Bender, E.A., Canfield, E.R.: The asymptotic number of labelled graphs with given degree sequence. Journal of Combinatorial Theory (A) 24, 296–307 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Chvatal, V., Szemeredi, E.: Many hard examples for resolution. Journal of the ACM 35, 759–768 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Creignou, N., Daude, H.: Random generalized satisfiability problems. In: Proceedings of SAT (2002)Google Scholar
  12. 12.
    Dechter, R.: Constraint networks. In: Shapiro, S. (ed.) Encyclopedia of Artificial Intelligence, 2nd edn., pp. 276–285. Wiley, New York (1992)Google Scholar
  13. 13.
    Dyer, M., Frieze, A., Molloy, M.: A probabilistic analysis of randomly generated binary constraint satisfaction problems. Theoretical Computer Scince 290, 1815–1828 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Freuder, E.C.: A sufficient condition for backtrack-free search. Journal of the ACM 29, 24–32 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bobrow, D.G., Brady, M. (eds.): Special Volume on Frontiers in Problem Solving: Phase Transitions and Complexity; Hogg, T., Hubermann, B.A., Williams, C.( Guest eds.): Artificial Intelligence 81(1-2) (1996)Google Scholar
  16. 16.
    Gent, I., MacIntyre, E., Prosser, P., Smith, B., Walsh, T.: Random constraint satisfaction: flaws and structure. Constraints 6, 345–372 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley, Chichester (2000)zbMATHGoogle Scholar
  18. 18.
    Mackworth, A.K.: Constraint satisfaction. In: Shapiro, S. (ed.) Encyclopedia of Artificial Intelligence, 2nd edn., pp. 285–293. Wiley, New York (1992)Google Scholar
  19. 19.
    Mitchell, D.: The Resolution complexity of random constraints. In: Van Hentenryck, P. (ed.) CP (2002)Google Scholar
  20. 20.
    Mitchell, D.: The Resolution Complexity of Constraint Satisfaction. Ph.D. Thesis, University of Toronto (2002)Google Scholar
  21. 21.
    Molloy, M.: Models for Random Constraint Satisfaction Problems. In: Proceedings of STOC 2002, pp. 209–217 (2002); Longer version to appear in SIAM J. ComputingGoogle Scholar
  22. 22.
    Molloy, M.: When does the giant component bring unsatisfiability? (submitted)Google Scholar
  23. 23.
    Molloy, M., Salavatipour, M.: The resolution complexity of random constraint satisfaction problems (submitted)Google Scholar
  24. 24.
    Pittel, B., Spencer, J., Wormald, N.: Sudden emergence of a giant k-core in a random graph. Journal of Combinatorial Theory (B) 67, 111–151 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Talagrand, M.: Concentration of mesure and isoperimetric inequalities. Inst. Hautes Études Sci. Publ. Math. 81, 73–205 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Waltz, D.: Understanding line drawings of scenes with shadows. In: Waltz, D. (ed.) The Psychology of Computer Vision, pp. 19–91. McGraw-Hill, New York (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alan Frieze
    • 1
  • Michael Molloy
    • 2
    • 3
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer ScienceUniversity of TorontoToronto
  3. 3.Microsoft ResearchRedmond

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