Random constraint satisfaction: A more accurate picture

  • Dimitris Achlioptas
  • Lefteris M. Kirousis
  • Evangelos Kranakis
  • Danny Krizanc
  • Michael S. O. Molloy
  • Yannis C. Stamatiou
Session 2b
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1330)


Recently there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Rather intriguingly, experimental results with various models for generating random CSP instances suggest a “threshold-like” behavior and some theoretical work has been done in analyzing these models when the number of variables becomes large (asymptotic). In this paper we prove that the models commonly used for generating random CSP instances do not have an asymptotic threshold. In particular, we prove that as the number of variables becomes large, almost add instances they generate are trivially overconstrained. We then present a new model for random CSP and, in the spirit of random k-SAT, we derive lower and upper bounds for its parameters so that instances are “almost surely” underconstrained and overconstrained, respectively. Finally, for the case of one of the popular models in Artificial Intelligence we derive sharper estimates for the probability of being overconstrained, as a function of the number of variables. Canada PGS B Scholarship. E-mail: optas@cs.toronto.edu


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Achlioptas and M. Molloy, “Almost all graphs with 2.522 n edges are not 3-colorable,” Manuscript, University of Toronto.Google Scholar
  2. 2.
    N. Alon, J.H. Spencer, and P. Erdös, The Probabilistic Method, J. Wiley, New York (1992).Google Scholar
  3. 3.
    D.G. Bobrow and M. Brady, eds., Special Volume on Frontiers in Problem Solving: Phase Transitions and Complexity, Guest editors: T. Hogg, B.A. Hubermann, and C.P. Williams, Artificial Intelligence, Vol. 81, Numbers 1–2 (1996).Google Scholar
  4. 4.
    B. Bollobás, Random Graphs, Academic Press, London (1985).Google Scholar
  5. 5.
    A.Z. Broder, A.M. Frieze, and E. Upfal, “On the satisfiability and maximum satisfiability of random 3-CNF formulas,” Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (1993) 322‐330.Google Scholar
  6. 6.
    M.-T. Chao and J. Franco, “Probabilistic analysis of two heuristics for the 3-satisfiability problem,” SIAM Journal on Computing, Vol. 15 (1986) 1106–1118.Google Scholar
  7. 7.
    M.-T. Chao and J. Franco, “Probabilistic analysis of a generalization of the unitclause literal selection heuristic for the k-satisfiability problem,” Information Science, Vol. 51 (1990) 289–314.Google Scholar
  8. 8.
    P. Cheeseman, B. Kanefsky and W. Taylor, “Where the really hard problems are,” Proceedings of IJCAI '91 (1991) 331–337.Google Scholar
  9. 9.
    V. Chvátal and B. Reed, “Mick gets some (the odds are on his side),” in Proceedings of 33rd IEEE Symposium on Foundations of Computer Science (1992) 620–627.Google Scholar
  10. 10.
    R. Dechter, “Constraint networks,” in S. Shapiro (ed.), Encyclopedia of Artificial Intelligence, Wiley, New York, 2nd ed. (1992) 276–285.Google Scholar
  11. 11.
    O. Dubois and Y. Boufkhad, A General Upper Bound for the Satisfiability Threshold of Random r-SAT Formulae, Preprint, LAFORIA, CNRS-Université Paris 6, 1996.Google Scholar
  12. 12.
    P. Erdös and A. Rényi, “On the evolution of random graphs,” Publ. of the Math. Inst. of the Hung. Acad. Sci., Vol. 5 (1960) 17–61.Google Scholar
  13. 13.
    A. El Maftouhi and W. Fernandez de la Vega, “On random 3-SAT”, Combinatorics, Probability and Computing, Vol. 4 (1995) 189–195.Google Scholar
  14. 14.
    E. Friedgut, “Sharp thresholds for graph properties and the k-sat problem,” Manuscript in preparation.Google Scholar
  15. 15.
    A. Frieze and S. Suen, “Analysis of two simple heuristics on a random instance of k-SAT,” Journal of Algorithms, Vol. 20 (1996) 312–355.Google Scholar
  16. 16.
    A. Goerdt, “A threshold for satisfiability,” in I.M. Haven and V. Koubek (eds.) Proceedings of the Symposium on the Mathematical Foundations of Computer Science (1992) 264–274.Google Scholar
  17. 17.
    T. Hogg, “Refining the phase transition in combinatorial search,” in [3], 127–154.Google Scholar
  18. 18.
    J. Kahn, G. Kalai, and N. Linial, “The influence of variables on Boolean functions”, Proceedings of the 29th Annual Symposium on the Foundations of Computer Science (1988) 68–80.Google Scholar
  19. 19.
    A. Kamath, R. Motwani, K. Palem, and P. Spirakis, “Tail bounds for occupancy and the satisfiability threshold conjecture,” Random Structures and Algorithms, Vol. 7 (1995) 59–80.Google Scholar
  20. 20.
    S. Kirkpatrick and B. Selman, “Critical behavior in the satisfiability of random Boolean expressions,” Science 264, pp 1297–1301, 1994.Google Scholar
  21. 21.
    L. M. Kirousis, E. Kranakis, and D. Krizanc, “Approximating the unsatisfiability threshold of random formulas,” Proceedings of the Fourth Annual European Symposium on Algorithms, ESA '96, (1996) 27–38.Google Scholar
  22. 22.
    A.K. Mackworth, “Constraint satisfaction,” in S. Shapiro (ed.) Encyclopedia of Artificial Intelligence, Wiley, New York, 2nd ed. (1992) 285–293.Google Scholar
  23. 23.
    C. McDiarmid, “On a correlation inequality of Farr,” Combinatorics, Probability and Computing, Vol. 1 (1992) 157–160.Google Scholar
  24. 24.
    D. Mitchell, B. Selman, and H. Levesque, Generating hard satisfaability problems, Artificial Intelligence, Vol. 81 (1996), 17–29.Google Scholar
  25. 25.
    P. Prosser, “An empirical study of phase transitions in binary constraint satisfaction problems,” in [3], 81–109.Google Scholar
  26. 26.
    B.M. Smith and M.E. Dyer, “Locating the phase transition in binary constraint satisfaction problems,” in [3], 155–181.Google Scholar
  27. 27.
    J. H. Spencer, Ten Lectures on the Probabilistic Method, 2nd edition, SIAM, Philadelphia (1994).Google Scholar
  28. 28.
    D. Waltz, “Understanding line drawings of scenes with shadows,” The Psychology of Computer Vision, McGraw-Hill, New York (1975) 19–91.Google Scholar
  29. 29.
    C. Williams and T. Hogg, “Using deep structure to locate hard problems,” Proceedings of AAAI-92 (1992) 472–477.Google Scholar
  30. 30.
    C. Williams and T. Hogg, “Extending deep structure,” Proceedings of AAAI-93, (1993) 152–158.Google Scholar
  31. 31.
    C. Williams and T. Hogg, “Exploiting the deep structure of constraint problems,” Artificial Intelligence, Vol. 70 (1994) 73–117.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Dimitris Achlioptas
    • 1
  • Lefteris M. Kirousis
    • 2
  • Evangelos Kranakis
    • 3
  • Danny Krizanc
    • 3
  • Michael S. O. Molloy
    • 1
  • Yannis C. Stamatiou
    • 2
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
  2. 2.Department of Computer Engineering and InformaticsUniversity of PatrasPatrasGreece
  3. 3.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations