Phase Transitions and Backbones of 3-SAT and Maximum 3-SAT

  • Weixiong Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2239)


Many real-world problems involve constraints that cannot be all satisfied. Solving an overconstrained problem then means to find solutions minimizing the number of constraints violated, which is an optimization problem. In this research, we study the behavior of the phase transitions and backbones of constraint optimization problems. We first investigate the relationship between the phase transitions of Boolean satis fiability, or precisely 3-SAT (a well-studied NP-complete decision problem), and the phase transitions of MAX 3-SAT (an NP-hard optimization problem). To bridge the gap between the easy-hard-easy phase transitions of 3-SAT and the easy-hard transitions of MAX 3-SAT, we analyze bounded 3-SAT, in which solutions of bounded quality, e.g., solutions with at most a constant number of constraints violated, are sufficient. We show that phase transitions are persistent in bounded 3-SAT and are similar to that of 3-SAT. We then study backbones of MAX 3-SAT, which are critically constrained variables that have fixed values in all optimal solutions. Our experimental results show that backbones of MAX 3-SAT emerge abruptly and experience sharp transitions from nonexistence when underconstrained to almost complete when overconstrained. More interestingly, the phase transitions of MAX 3-SAT backbones seem to concur with the phase transitions of satisfiability of 3-SAT. The backbone of MAX 3-SAT with size 0.5 approximately collocates with the 0.5 satisfiability of 3-SAT, and the backbone and satisfiability seems to follow a linear correlation near this 0.5-0.5 collocation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Achlioptas, C. Gomes, H. Kautz, and B. Selman. Generating satisfiable problem instances. In Proceedings of the 17th National Conference on Artificial Intelligence (AAAI-00), pages 256–261, Austin, Texas, July–August 2000.Google Scholar
  2. 2.
    J. C. Beck and M. S. Fox. A generic framework for constraint-directed search and scheduling. AI Magazine, 19(4):101–130, 1998.Google Scholar
  3. 3.
    P. Cheeseman, B. Kanefsky, and W. M. Taylor. Where the really hard problems are. In Proceedings of the 12th International Joint Conference on Artificial Intelligence, (IJCAI-91), pages 331–337, Sydney, Australia, August 1991.Google Scholar
  4. 4.
    P. Codognet and F. Rossi. Notes for the ECAI2000 tutorial on Solving and Programming with Soft Constraints: Theory and Practice. Available at
  5. 5.
    Joseph Culberson and Ian P. Gent. Frozen development in graph coloring. Theoretical Computer Science, page to appear, 2001.Google Scholar
  6. 6.
    M. Davis, G. Logemann, and D. Loveland. A machine program for theorem proving. Communications of ACM, 5:394–397, 1962.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    E. C. Freuder and R. J. Wallace. Partial constraint satisfaction. Artificial Intelligence, 58:21–70, 1992.CrossRefMathSciNetGoogle Scholar
  8. 8.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, NY, 1979.zbMATHGoogle Scholar
  9. 9.
    I. Gent and T. Walsh. Phase transitions and annealed theories: Number partitioning as a case stud,. In ECAI-96, pages 170–174, 1996.Google Scholar
  10. 10.
    I. P. Gent and T. Walsh. The TSP phase transition. Artificial Intelligence, 88:349–358, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    T. Hogg, B. A. Huberman, and C. Williams. Phase transitions and the search problem. Artificial Intelligence, 81:1–15, 1996.CrossRefMathSciNetGoogle Scholar
  12. 12.
    B. A. Huberman and T. Hogg. Phase transitions in artificial intelligence systems. Artificial Intelligence, 33:155–171, 1987.CrossRefGoogle Scholar
  13. 13.
    R. M. Karp and J. Pearl. Searching for an optimal path in a tree with random costs. Artificial Intelligence, 21:99–117, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Kirkpatrick and G. Toulouse. Configuration space analysis of traveling salesman problems. J. de Physique, 46:1277–1292, 1985.MathSciNetCrossRefGoogle Scholar
  15. 15.
    C. J. H. McDiarmid. Probabilistic analysis of tree search. In G. R. Gummett and D. J. A. Welsh, editors, Disorder in Physical Systems, pages 249–260. Oxford Science, 1990.Google Scholar
  16. 16.
    D. Mitchell, B. Selman, and H. Levesque. Hard and easy distributions of SAT problems. In Proceedings of the 10th National Conference on Artificial Intelligence (AAAI-92), pages 459–465, San Jose, CA, July 1992.Google Scholar
  17. 17.
    R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky. Determining computational complexity from characteristic ‘phase transitions’. Nature, 400:133–137, 1999.CrossRefMathSciNetGoogle Scholar
  18. 18.
    A. J. Parkes. Clustering at the phase transition. In Proceedings of the 14th National Conference on Artificial Intelligence (AAAI-97), pages 340–245, Providence, RI, July, 1997.Google Scholar
  19. 19.
    J. Singer, I. P. Gent, and A. Smaill. Backbone fragility and the local search cost peak. J. Artificial Intelligence Research, 12:235–270, 2000.zbMATHMathSciNetGoogle Scholar
  20. 20.
    J. Slaney and S. Thiebaux. On the hardness of decision and optimisation problems. In Proceedings of ECAI-98, pages 224–248, 1998.Google Scholar
  21. 21.
    J. Slaney and T. Walsh. Backbones in optimization and approximation. In Proceedings of the 17th International Joint Conference on Artificial Intelligence, (IJCAI-01), page to appear, Seattle, WA, August 2001.Google Scholar
  22. 22.
    E. Tsang. Foundations of Constraint Satisfaction. Academic Press, London, 1993.Google Scholar
  23. 23.
    W. Zhang. State-Space Search: Algorithms, Complexity, Extensions, and Applications. Springer, New York, NY, 1999.zbMATHGoogle Scholar
  24. 24.
    W. Zhang and R. E. Korf. Performance of linear-space search algorithms. Artificial Intelligence, 79:241–292, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    W. Zhang and R. E. Korf. A study of complexity transitions on the asymmetric Traveling Salesman Problem. Artificial Intelligence, 81:223–239, 1996.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Weixiong Zhang
    • 1
  1. 1.Department of Computer ScienceWashington UniversitySt.Louis

Personalised recommendations