computational complexity

, Volume 20, Issue 4, pp 649–678 | Cite as

Satisfiability, Branch-Width and Tseitin tautologies

  • Michael Alekhnovich
  • Alexander Razborov


For a CNF τ, let w b (τ) be the branch-width of its underlying hypergraph, that is the smallest w for which the clauses of τ can be arranged in the form of leaves of a rooted binary tree in such a way that for every vertex its descendants and non-descendants have at most w variables in common. In this paper we design an algorithm for solving SAT in time \({n^{O(1)}2^{O(w_b(\tau))}}\). This in particular implies a polynomial algorithm for testing satisfiability on instances with branch-width O(log n). Our algorithm is a modification of the width based automated theorem prover (WBATP) which is a popular (at least on the theoretical level) heuristic for finding resolution refutations of unsatisfiable CNFs, and we call it Branch-Width Based Automated Theorem Prover (BWBATP). As opposed to WBATP, our algorithm always produces regular refutations. Perhaps more importantly, its running time is bounded in terms of a clean combinatorial characteristic that can be efficiently approximated, and that the algorithm also produces, within the same time, a satisfying assignment if τ happens to be satisfiable.

In the second part of the paper we investigate the behavior of BWBATP on the well-studied class of Tseitin tautologies. We argue that in this case BWBATP is better than WBATP. Namely, we show that its running time on any Tseitin tautology τ is \({|\tau|^{O(1)} \cdot 2^{O(w(\tau\vdash\emptyset))}}\) , as opposed to the obvious bound \({n^{O(w(\tau\vdash\emptyset))}}\) provided by WBATP.

This in particular implies that resolution is automatizable on those Tseitin tautologies for which we know the relation \({w(\tau\vdash\emptyset)\leq O(\log S(\tau))}\). We identify one such subclass and prove partial results toward establishing this relation for larger classes of graphs.


SAT provers Automatizability Branch-width 

Subject classification

68Q17 68Q25 68T15 03F20 


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  1. M. Alekhnovich (2004). Mutilated chessboard problem is exponentially hard for resolution. Theoretical Computer Science 310(1-3), 513–525.Google Scholar
  2. Alekhnovich M., Ben-Sasson E., Razborov A., Wigderson A. (2004) Pseudorandom generators in propositional proof complexity. SIAM Journal on Computing 34(1): 67–88zbMATHCrossRefMathSciNetGoogle Scholar
  3. M. Alekhnovich & A. Razborov (2003). Lower bounds for the polynomial calculus: non-binomial case. Proceedings of the Steklov Institute of Mathematics 242, 18-35.Google Scholar
  4. Alekhnovich M., Razborov A. (2008) Resolution is not automatizable unless W[P] is tractable. SIAM Journal on Computing 38(4): 1347–1363zbMATHCrossRefMathSciNetGoogle Scholar
  5. Alon N., Boppana R. (1987) The monotone circuit complexity of Boolean functions. Combinatorica 7(1): 1–22zbMATHCrossRefMathSciNetGoogle Scholar
  6. E. Amir & S. McIlraith (2001). Solving Satisfiability using Decomposition and the Most Constrained Subproblem. In LICS workshop on Theory and Applications of Satisfiability Testing (SAT 2001).Google Scholar
  7. Atserias A., Bonet M. (2004) On the Automatizability of Resolution and Related Propositional Proof Systems. Information and Computation 189(2): 182–201zbMATHCrossRefMathSciNetGoogle Scholar
  8. P. Beame & T. Pitassi (1996). Simplified and improved resolution lower bounds. In Proceedings of the 37th IEEE FOCS, 274–282.Google Scholar
  9. E. Ben-Sasson & R. Impagliazzo (1999). Random CNF’ s are Hard for the Polynomial Calculus. In Proceedings of the 40th IEEE FOCS, 415–421.Google Scholar
  10. Ben-Sasson E., Wigderson A. (2001) Short Proofs are Narrow - Resolution made Simple. Journal of the ACM 48(2): 149–169zbMATHCrossRefMathSciNetGoogle Scholar
  11. Bodlaender H.L. (1993) A Tourist Guide through Treewidth. Acta Cybernetica 11: 1–21zbMATHMathSciNetGoogle Scholar
  12. M. Bonet & N. Galesi (1999). A study of proof search algorithms for Resolution and Polynomial Calculus. In Proceedings of the 40th IEEE FOCS, 422–431.Google Scholar
  13. Bonet M., Pitassi T., Raz R. (2000) On Interpolation and Automatization for Frege Systems. SIAM Journal on Computing 29(6): 1939–1967zbMATHCrossRefMathSciNetGoogle Scholar
  14. S. Chen, T. Lou, P. Papakonstantinou & B. Tang (2011). Width-parameterized SAT: Time-Space Tradeoffs. Technical Report cs.CC/1108.2385, arXiv e-print.Google Scholar
  15. B. Courcelle, J. A. Makowsky & U. Rotics (2001). On the Fixed Parameter Complexity of Graph Enumeration Problems Definable in Monadic Second Order Logic. Discrete Applied Mathematics 108(1-2), 23–52.Google Scholar
  16. Dantsin E. (1979) Parameters defining the time of tautology recognition by the splitting method. Semiotics and information science 12: 8–17 In RussianMathSciNetGoogle Scholar
  17. Erdös P., Rado R. (1960) Intersection theorems for systems of sets. Journal of the London Math. Society 35: 85–90zbMATHCrossRefGoogle Scholar
  18. S. Khanna & R. Motwani (1996). Towards a Syntactic Characterization of PTAS. In Proceedings of the 28th ACM Symposium on the Theory of Computing, 329–337.Google Scholar
  19. J. Krajíček (1992). No counter-example interpretation and interactive computation. In Logic from Computer Science, Y. N. Moschovakis, editor, 287–293. Springer-Verlag.Google Scholar
  20. Lipton R., Tarjan R. (1979) A Separator Theorem for Planar Graphs. SIAM Journal on Applied Mathematics 36: 177–189zbMATHCrossRefMathSciNetGoogle Scholar
  21. Lipton R., Tarjan R. (1980) Applications of a Planar Separator Theorem. SIAM Journal on Computing 9: 615–627zbMATHCrossRefMathSciNetGoogle Scholar
  22. A. A. Razborov (1985). Lower bounds for the monotone complexity of some Boolean functions. Doklady Academii Nauk SSSR 281(4), 798–801. English Translation in Soviet Math. Dokl., 31:354-357, 1985.Google Scholar
  23. N. Robertson & P. D. Seymour (1991). Graph minors. X. Obstructions to tree decomposition. Journal of Combinatorial Theory Series B 52, 153–190.Google Scholar
  24. Robertson N., Seymour P.D. (1995) Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory Series B 63: 65–110zbMATHCrossRefMathSciNetGoogle Scholar
  25. G. S. Tseitin (1968). On the complexity of derivations in propositional calculus. In Studies in constructive mathematics and mathematical logic, Part II. Consultants Bureau, New-York-London.Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Laboratory for Computer ScienceMITCambridgeUSA
  2. 2.Institute for Advanced StudyPrincetonUSA
  3. 3.University of ChicagoChicagoUSA
  4. 4.Steklov Mathematical InstituteMoscowRussia

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