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

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Computing and Combinatorics (COCOON 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2697))

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Abstract

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.

Supported by the Austrian Science Fund (FWF) Project J2111.

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References

  1. R. Aharoni and N. Linial. Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Combin. Theory Ser. A, 43:196–204, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  5. R. Diestel. Graph Theory. Springer Verlag, New York, 2nd edition, 2000.

    Google Scholar 

  6. R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer Verlag, 1999.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Franco and A. Van Gelder. A perspective on certain polynomial time solvable classes of satisfiability. Discr. Appl. Math., 125:177–214, 2003.

    Article  MATH  Google Scholar 

  9. G. Gottlob, F. Scarcello, and M. Sideri. Fixed-parameter complexity in AI and nonmonotonic reasoning. Artificial Intelligence, 138(1–2):55–86, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  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. H. Kleine Büning. On subclasses of minimal unsatisfiable formulas. Discr. Appl. Math., 107(1–3):83–98, 2000.

    Article  MATH  Google Scholar 

  12. O. Kullmann. Lean clause-sets: Generalizations of minimally unsatisfiable clausesets. To appear in Discr. Appl. Math.

    Google Scholar 

  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. L. Lovász and M. D. Plummer. Matching Theory. North-Holland Publishing Co., Amsterdam, 1986.

    MATH  Google Scholar 

  15. B. Monien and E. Speckenmeyer. Solving satisfiability in less than 2n steps. Discr. Appl. Math., 10:287–295, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. H. Papadimitriou and D. Wolfe. The complexity of facets resolved. J. Comput. System Sci., 37(1):2–13, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  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. C. A. Tovey. A simplified NP-complete satisfiability problem. Discr. Appl. Math., 8(1):85–89, 1984.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Toronto, M5S 3G4, Toronto, Ontario, Canada

    Stefan Szeider

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  1. Stefan Szeider
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Texas at Austin, One University Station, C0500, Austin, TX, 78712, USA

    Tandy Warnow

  2. Department of Computer Science, Montana State University, EPS 357, Bozeman, MT, 59717, USA

    Binhai Zhu

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Szeider, S. (2003). Minimal Unsatisfiable Formulas with Bounded Clause-Variable Difference are Fixed-Parameter Tractable. In: Warnow, T., Zhu, B. (eds) Computing and Combinatorics. COCOON 2003. Lecture Notes in Computer Science, vol 2697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45071-8_55

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  • DOI: https://doi.org/10.1007/3-540-45071-8_55

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40534-4

  • Online ISBN: 978-3-540-45071-9

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