The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

  • Parikshit Gopalan
  • Phokion G. Kolaitis
  • Elitza N. Maneva
  • Christos H. Papadimitriou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)


Given a Boolean formula, do its solutions form a connected subgraph of the hypercube? This and other related connectivity considerations underlie recent work on random Boolean satisfiability. We study connectivity properties of the space of solutions of Boolean formulas, and establish computational and structural dichotomies. Specifically, we first establish a dichotomy theorem for the complexity of the st-connectivity problem for Boolean formulas in Schaefer’s framework. Our result asserts that the tractable side is more generous than the tractable side in Schaefer’s dichotomy theorem for satisfiability, while the intractable side is PSPACE-complete. For the connectivity problem, we establish a dichotomy along the same boundary between membership in coNP and PSPACE-completeness. Furthermore, we establish a structural dichotomy theorem for the diameter of the connected components of the solution space: for the PSPACE-complete cases, the diameter can be exponential, but in all other cases it is linear. Thus, small diameter and tractability of the st-connectivity problem are remarkably aligned.


Constraint Satisfaction Problem Dichotomy Theorem Logical Relation Boolean Formula Connectivity Property 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Schaefer, T.: The complexity of satisfiability problems. In: Proc. 10th ACM Symp. Theory of Computing, pp. 216–226 (1978)Google Scholar
  2. 2.
    Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. Journal of Computer and System Sciences 51, 511–522 (1995)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Creignou, N., Khanna, S., Sudan, M.: Complexity classification of Boolean constraint satisfaction problems. In: SIAM Monographs on Disc. Math., vol. 7, SIAM, Philadelphia (2001)Google Scholar
  4. 4.
    Khanna, S., Sudan, M., Trevisan, L., Williamson, D.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Information and Computation 125(1), 1–12 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kavvadias, D., Sideri, M.: The inverse satisfiability problem. SIAM J. Comput. 28(1), 152–163 (1998)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kirousis, L., Kolaitis, P.: The complexity of minimal satisfiability problems. Information and Computation 187(1), 20–39 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bulatov, A.: A dichotomy theorem for constraints on a three-element set. In: Proc. 43rd IEEE Symp. Foundations of Computer Science, pp. 649–658 (2002)Google Scholar
  9. 9.
    Creignou, N., Zanuttini, B.: A complete classification of the complexity of propositional abduction. SIAM Journal on Computing (to appear, 2006)Google Scholar
  10. 10.
    Achlioptas, D., Naor, A., Peres, Y.: Rigorous location of phase transitions in hard optimization problems. Nature 435, 759–764 (2005)CrossRefGoogle Scholar
  11. 11.
    Mézard, M., Zecchina, R.: Random k-satisfiability: from an analytic solution to an efficient algorithm. Phys. Rev. E 66 (2002)Google Scholar
  12. 12.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812 (2002)Google Scholar
  13. 13.
    Maneva, E., Mossel, E., Wainwright, M.J.: A new look at survey propagation and its generalizations. In: Proc. 16th ACM-SIAM Symp. Discrete Algorithms, 1089–1098, pp. 1089–1098 (2005)Google Scholar
  14. 14.
    Mora, T., Mézard, M., Zecchina, R.: Clustering of solutions in the random satisfiability problem. Phys. Rev. Lett. ( in press, 2005)Google Scholar
  15. 15.
    Achlioptas, D., Ricci-Tersenghi, F.: On the solution-space geometry of random constraint satisfaction problems. In: 38th ACM Symp. Theory of Computing (2006)Google Scholar
  16. 16.
    Selman, B., Kautz, H., Cohen, B.: Local search strategies for satisfiability testing. In: Cliques, coloring, and satisfiability: second DIMACS implementation challenge, October 1993, AMS (1996)Google Scholar
  17. 17.
    Achlioptas, D., Beame, P., Molloy, M.: Exponential bounds for DPLL below the satisfiability threshold. In: Proc. 15th ACM-SIAM Symp. Discrete Algorithms, pp. 132–133 (2004)Google Scholar
  18. 18.
    Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with Boolean blocks, Part II: constraint satisfaction problems. ACM SIGACT-Newsletter 35(1), 22–35 (2004)CrossRefGoogle Scholar
  19. 19.
    Hearne, R., Demaine, E.: The Nondeterministic Constraint Logic model of computation: Reductions and applications. In: 29th Intl. Colloquium on Automata, Languages and Programming, pp. 401–413 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Parikshit Gopalan
    • 1
  • Phokion G. Kolaitis
    • 2
  • Elitza N. Maneva
    • 3
  • Christos H. Papadimitriou
    • 3
  1. 1.Georgia Tech 
  2. 2.IBM Almaden Research Center 
  3. 3.UC Berkeley 

Personalised recommendations