Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable

  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2697)


Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable by removing any clause) is NP-hard; it was shown recently that minimal unsatisfiable formulas with n variables and n + k clauses can be recognized in time n O(k). We improve this result and present an algorithm with time complexity O(2kn4) —hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, Springer Verlag, 1999).

Our algorithm gives rise to an FPT parameterization of SAT (“maximum deficiency”) which is incomparable with known FPT parameterizations of SAT like tree-width.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Aharoni and N. Linial. Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Combin. Theory Ser. A, 43:196–204, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Alekhnovich and A. A. Razborov. Satisfiability, branch-width and Tseitin tautologies. In Proceedings of the 43rd IEEE FOCS, pages 593–603, 2002.Google Scholar
  3. 3.
    B. Courcelle, J. A. Makowsky, and U. Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discr. Appl. Math., 108(1–2):23–52, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Davydov, I. Davydova, and H. Kleine Büning. An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Ann. Math. Artif. Intell., 23:229–245, 1998.zbMATHCrossRefGoogle Scholar
  5. 5.
    R. Diestel. Graph Theory. Springer Verlag, New York, 2nd edition, 2000.Google Scholar
  6. 6.
    R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer Verlag, 1999.Google Scholar
  7. 7.
    H. Fleischner, O. Kullmann, and S. Szeider. Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoret. Comput. Sci., 289(1):503–516, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Franco and A. Van Gelder. A perspective on certain polynomial time solvable classes of satisfiability. Discr. Appl. Math., 125:177–214, 2003.zbMATHCrossRefGoogle Scholar
  9. 9.
    G. Gottlob, F. Scarcello, and M. Sideri. Fixed-parameter complexity in AI and nonmonotonic reasoning. Artificial Intelligence, 138(1–2):55–86, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    H. Kleine Büning. An upper bound for minimal resolution refutations. In Proc. CSL’98, volume 1584 of LNCS, pages 171–178. Springer Verlag, 1999.Google Scholar
  11. 11.
    H. Kleine Büning. On subclasses of minimal unsatisfiable formulas. Discr. Appl. Math., 107(1–3):83–98, 2000.zbMATHCrossRefGoogle Scholar
  12. 12.
    O. Kullmann. Lean clause-sets: Generalizations of minimally unsatisfiable clausesets. To appear in Discr. Appl. Math. Google Scholar
  13. 13.
    O. Kullmann. An application of matroid theory to the SAT problem. In Fifteenth Annual IEEE Conference on Computational Complexity, pages 116–124, 2000.Google Scholar
  14. 14.
    L. Lovász and M. D. Plummer. Matching Theory. North-Holland Publishing Co., Amsterdam, 1986.zbMATHGoogle Scholar
  15. 15.
    B. Monien and E. Speckenmeyer. Solving satisfiability in less than 2n steps. Discr. Appl. Math., 10:287–295, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    C. H. Papadimitriou and D. Wolfe. The complexity of facets resolved. J. Comput. System Sci., 37(1):2–13, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Szeider. Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. Technical Report TR03-002, Revision 1, Electronic Colloquium on Computational Complexity (ECCC), 2003.Google Scholar
  18. 18.
    C. A. Tovey. A simplified NP-complete satisfiability problem. Discr. Appl. Math., 8(1):85–89, 1984.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Stefan Szeider
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations