Skip to main content
SpringerLink
Log in

Random Constraint Satisfaction: Flaws and Structure

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

A recent theoretical result by Achlioptas et al. shows that many models of random binary constraint satisfaction problems become trivially insoluble as problem size increases. This insolubility is partly due to the presence of ‘flawed variables,’ variables whose values are all ‘flawed’ (or unsupported). In this paper, we analyse how seriously existing work has been affected. We survey the literature to identify experimental studies that use models and parameters that may have been affected by flaws. We then estimate theoretically and measure experimentally the size at which flawed variables can be expected to occur. To eliminate flawed values and variables in the models currently used, we introduce a ‘flawless’ generator which puts a limited amount of structure into the conflict matrix. We prove that such flawless problems are not trivially insoluble for constraint tightnesses up to 1/2. We also prove that the standard models B and C do not suffer from flaws when the constraint tightness is less than the reciprocal of domain size. We consider introducing types of structure into the constraint graph which are rare in random graphs and present experimental results with such structured graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achlioptas, D., Kirousis, L. M., Kranakis, E., Krizanc, D., Molloy, M. S. O., & Stamatiou, Y. C. (1997). Random constraint satisfaction: A more accurate picture. In Proc. CP97, pages 107–120. Springer.

  2. Bacchus, F., & Grove, A. (1995). On the forward checking algorithm. In U. Montanari & F. Rossi, eds. Principles and Practice of Constraint Programming, pages 292–309. Springer.

  3. Bacchus, F., & van Run, P. (1995). Dynamic variable ordering in CSP's. In U. Montanari & F. Rossi, eds. Principles and Practice of Constraint Programming, pages 258–275. Springer.

  4. Bessière, C., & Régin, J.–C. (1996). MAC and combined heuristics: Two reasons to forsake FC (and CBJ?) on hard problems. In E. C. Freuder, ed. Proceedings of Second International Conference on Principles and Practice of Constraint Programming (CP96), pages 61–75. Springer.

  5. Borrett, J. E., Tsang, E., & Walsh, N. R. (1996). Adaptive constraint satisfaction: The quickest first principle. In W. Wahlster, ed. Proceedings of the 12th ECAI, pages 160–164. European Conference on Artificial Intelligence, Wiley.

  6. Cabon, B., Verfaillie, G., Martinez, D., & Bourret, P. (1996). Using mean field methods for boosting backtrack search in constraint satisfaction problems. In W. Wahlster, ed. Proceedings of the 12th ECAI, European Conference on Artificial Intelligence, pages 165–169. Wiley.

  7. Cheeseman, P., Kanefsky, B., & Taylor, W. M. (1991). Where the really hard problems are. In Proceedings of the 12th IJCAI, pages 331–337.

  8. Chmeiss A., & Jégou, P. (1996). Path–consistency: When space misses time. In Proceedings of AAAI–96, pages 196–201.

  9. Clark, D. A., Frank, J., Gent, I. P., MacIntyre, E., Tomov, N., & Walsh, T. (1996). Local search and the number of solutions. In Proceedings of CP–96, pages 119–133. Springer.

  10. Debruyne, R., & Bessière, C. (1997). From restricted path consistency to max–restricted path consistency. In G. Smolka, ed. Proceedings of Third International Conference on Principles and Practice of Constraint Programming (CP97), pages 312–326. Springer.

  11. Debruyne, R., & Bessière, C. (1997). Some practicable filtering techniques for the constraint satisfaction problem. In Proceedings of the 15th IJCAI, pages 412–417. International Joint Conference on Artificial Intelligence.

  12. Freuder, E. (1982). A sufficient condition for backtrack–free search. Journal of the Association for Computing Machinery, 29(1): 24–32.

    Google Scholar 

  13. Freuder E. C., & Elfe, C. D. (1996). Neighbourhood inverse consistency preprocessing. In Proceedings of AAAI–96, pages 202–208.

  14. Freuder, E. C., & Hubbe, P. D. (1995). Extracting constraint satisfaction subproblems. In Proceedings of the 14th IJCAI, pages 548–555. International Joint Conference on Artificial Intelligence.

  15. Friedgut, E. (1998). Sharp thresholds for graph properties and the k–SAT problem. Unpublished manuscript.

  16. Frieze, A., & Suen, S. (1996). Analysis of two simple heuristics on a random instance of k–SAT. Journal of Algorithms. 20: 312–355.

    Google Scholar 

  17. Frost, D., & Dechter, R. (1994). Dead–end driven learning. In Proceedings AAAI–94, pages 294–300.

  18. Frost, D., & Dechter, R. (1994). In search of the best search: An empirical evaluation. In Proceedings AAAI–94, pages 301–306.

  19. Frost, D., & Dechter, R. (1995). Look–ahead value ordering for constraint satisfaction problems. In Proceedings of the 14th IJCAI, pages 572–578. International Joint Conference on Artificial Intelligence.

  20. Frost, D., Rish, I., & Vila, L. (1997). Summarizing CSP hardness with continuous probability distributions. In Proceedings of the 14th National Conference on AI, pages 327–333. American Association for Artificial Intelligence.

  21. Galinier, P., & Hao, J.–K. (1997). Tabu search for maximal constraint satisfaction problems. In G. Smolka, editor, Proceedings of Third International Conference on Principles and Practice of Constraint Programming (CP97), pages 196–208. Springer.

  22. Gaur, D. R., Jackson, W. K., & Havens, W. S. (1997). Detecting unsatisfiable CSPs by coloring the micro–structure. In Proceedings of the 14th National Conference on AI, pages 215–220. American Association for Artificial Intelligence.

  23. Gent, I. P., MacIntyre, E., Prosser, P., Shaw, P., & Walsh, T. (1997). The constrainedness of are consistency. In Proceedings of CP–97 pages 327–340. Springer.

  24. Gent, P., MacIntyre, E., Professor, P., Smith, B. M., & Walsh, T. (1996). An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In Proceedings of CP–96, pages 179–193. Springer.

  25. Gent, I. P., MacIntyre, E., Prosser, P., Smith, B. M., & Walsh, T. (1998). Random constraint satisfaction: Flaws and structure. Technical Report APES–08–1998, APES Research Group.

  26. Gent, I. P., MacIntyre, E., Prosser, P., & Walsh, T. (1995). Scaling effects in the CSP phase transition. In U. Montanari and F. Rossi, ed. Principles and Practice of Constraint Programming. pages 70–87. Springer.

  27. Gent, I. P., MacIntyre, E., Prossor, P., & Walsh, T. (1996). The constrainedness of search. In Proceedings of AAAI–96, pages 246–252.

  28. Gent, I. P., MacIntyre, E., Prosser, P., & Walsh, T. (1997). The scaling of search cost. In Proceedings of AAAI–97, pages 315–320.

  29. Gent, I. P., & Underwood, J. (1997). The logic of search algorithms: Theory and applications. In Principles and Practice of Constraint Programming—CP97, pages 77–91. Springer.

  30. Gent, I. P., & Walsh, T. (1995). Phase transitions from real computational problems. In Proceedings of the 8th International Symposium on Artificial Intelligence, pages 356–364.

  31. Gent, I. P., & Walsh, T. (1998). Analysis of heuristics for number partitioning. Computational Intelligence, pages 430–451.

  32. Gomes, C., & Selman, B. (1997). Problem structure in the presense of perturbations. In Proceedings of the 14th National Conference on AI, pages 221–226. American Association for Artificial Intelligence.

  33. Grant, S., & Smith, B. M. (1996). The phase transition behaviour of maintaining are consistency, Research Report 95.25, School of Computer Studies, University of Leeds, 1995. A revised and shortened version appears in Proceedings of 12th ECAI, pages 175–179.

  34. Grant, S. A., & Smith, B. M. (1996). The phase transition behaviour of maintaining are consistency. In Proceedings of ECAI–96, pages 175–179.

  35. Kask, K., & Dechter, R. (1995). Gsat and local consistency. In Proceedings of the 14th IJCAI, pages 616–622. International Joint Conference on Artificial Intelligence.

  36. Kask, K., & Dechter, R. (1996). A graph–based method for improving GSAT. In Proceedings of AAAI–96, pages 350–355.

  37. Kirousis, L. M., Kranakis, E., & Krizanc, D. (1996). Approximating the unsatisfiability threshold of random formulas. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA'96), pages 27–38.

  38. Kwan, A. C. M., Tsang, E. P. K., & Borrett, J. E. (1996). Predicting phase transitions of binary CSPs with constraint graph information. In W. Wahlster, editor, Proceedings of the 12th ECAI, pages 185–189. European Conference on Artificial Intelligence, Wiley.

  39. Larrosa, J., & Meseguer, P. (1995). Optimization–based heuristics for maximal constraint satisfaction. In Proceedings of Second International Conference on Principles and Practice of Constraint Programming (CP95), pages 103–120. Springer.

  40. Larrosa J., & Meseguer, P. (1996). Exploiting the use of DAC in MAX–CSP. In E. C. Freuder, ed. Proceedings of Second International Conference on Principles and Practice of Constraint Programming (CP96), pages 308–322. Springer.

  41. Larrosa J., & Meseguer, P. (1996). Phase transition in MAX–CSP. In W. Wahlster, ed. Proceedings of the 12th ECAI, pages 190–194. European Conference on Artificial Intelligence, Wiley.

  42. Lesaint, D. (1994). Maximal sets of solutions for constraint satisfaction problems. In Proceedings of ECAI–94, pages 110–114.

  43. Meisels, A., Shimony, S. E., & Solotorevsky, G. (1997). Bayes networks for estimating the number of solutions to a CSP. In Proceedings of the 14th National Conference on AI, pages 179–184. American Association for Artificial Intelligence.

  44. Meseguer, P., & Larrosa, J. (1995). Constraint satisfaction as global optimization. In Proceedings of the 14th IJCAI, pages 579–584. International Joint Conference on Artificial Intelligence.

  45. Mitchell, D., Selman, B., & Levesque, H. (1992). Hard and easy distributions of SAT problems. In Proceedings of AAAI–92, pages 459–465.

  46. Palmer, E. M. (1985). Graphical Evolution: An introduction to the Theory of Random Graphs. John Wiley and Sons.

  47. Prosser, P. (1994). Binary constraint satisfaction problmes: Some are harder than others. In Proceedings of ECAI–94, pages 95–99.

  48. Rish, I., & Frost, D. (1997). Statistical analysis of backtracking on inconsistent CSPs. In G. Smolka, ed. Proceedings of Third International Conference on Principles and Practice of Constraint Programming (CP97), pages 150–162. Springer.

  49. Sabin, D., & Freuder, E. C. (1994). Contradicting conventional wisdom in constraint satisfaction. In Proceedings of ECAI–94, pages 125–129.

  50. Sabin, D., & Freuder, E. C. (1997). Understanding and improving the mac algorithm. In G. Smolka, ed. Proceedings of Third International Conference on Principles and Practice of Constraint Programming (CP97), pages 167–181. Springer.

  51. Smith, B. M. (1994). Phase transition and the mushy region in constraint satisfaction problems. In Proceedings of ECAI–94, pages 100–104.

  52. Smith, B. M., & Dyer, M. E. (1996). Locating the phase transition in binary constraint satisfaction problems. Artificial Intelligence, 81: 155–181.

    Google Scholar 

  53. Smith, B. M., & Grant, S. (1995). Where the exceptionally hard problems are. In Proceedings of the CP–95 Workshop on Studying and Solving Really Hard Problems, pages 172–182.

  54. Smith, B. M., & Grant, S. (1997). Modelling exceptionally hard constraint satisfaction problems. In G. Smolka, ed. Proceedings of Third International Conference on Principles and Practice of Constraint Programming (CP97), pages 182–195. Springer.

  55. Smith, B. M., & Grant S. A. (1995). Sparse constraint graphs and exceptionally hard problems. In Proceedings of IJCAI–95, pages 646–651.

  56. Smith B. M., Constructing an Asymptotic Phase Transition in Random Binary Constraint Satisfaction Problems. To appear in Theoretical Computer Science, Special Issue on NP–Hardness and Phase Transitions.

  57. Wallace, R. J. (1996). Analysis of heuristic methods for partial constraint satisfaction problems. In E. C. Freuder, editor, Proceedings of Second International Conference on Principles and Practice of Constraint Programming (CP96), pages 482–496. Springer.

  58. Williams, C., & Hogg, T. (1994). Exploiting the deep structure of constraint problems. Artificial, Intelligence, 70: 73–117.

    Google Scholar 

  59. Xu, K., & Li, W. (2000). Exact Phase Transitions in Random Constraint Satisfaction Problems. Journal of AI Research, 12: 93–103.

    Google Scholar 

  60. Yugami, N., Ohta, Y., & Hara, H. (1994). Improving repair–based constraint satisfaction methods by value propagation. In Proceedings AAAI–94, pages 344–349.

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, University of St Andrews, St Andrews, Fife, KY16 9SS, United Kingdom

    Ian P. Gent

  2. Broadwood Business Park, Cisco Systems (Scotland) Ltd., Cumbernauld, Glasgow, G68 9LE, United Kingdom

    Ewan Macintyre

  3. Department of Computing Science, University of Glasgow, Glasgow, G12 8QQ, United Kingdom

    Patrick Prosser

  4. School of Computing & Mathematics, University of Huddersfield, Huddersfield, HD1 3DH, United Kingdom

    Barbara M. Smith

  5. Department of Computer Science, University of York, York, YO10 5DD, United Kingdom

    Toby Walsh

Authors
  1. Ian P. Gent
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ewan Macintyre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Patrick Prosser
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Barbara M. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Toby Walsh
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gent, I.P., Macintyre, E., Prosser, P. et al. Random Constraint Satisfaction: Flaws and Structure. Constraints 6, 345–372 (2001). https://doi.org/10.1023/A:1011454308633

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011454308633

Search

Navigation