A New Bound for an NP-Hard Subclass of 3-SAT Using Backdoors

  • Stephan Kottler
  • Michael Kaufmann
  • Carsten Sinz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4996)


Knowing a Backdoor Set B for a given Sat instance, satisfiability can be decided by only examining each of the 2|B| truth assignments of the variables in B. However, one problem is to efficiently find a small backdoor up to a particular size and, furthermore, if no backdoor of the desired size could be found, there is in general no chance to conclude anything about satisfiability.

In this paper we introduce a complete deterministic algorithm for an NP-hard subclass of 3-Sat, that is also a subclass of Mixed Horn Formulas (MHF). For an instance of the described class the absence of two particular kinds of backdoor sets can be used to prove unsatisfiability. The upper bound of this algorithm is O(p(n)*1.427 n ) which is less than the currently best upper bound for deterministic algorithms for 3-Sat and MHF.


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  1. 1.
    Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Proc. Lett. 8, 121–123 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Brüggemann, T., Kern, W.: An improved deterministic local search algorithm for 3-sat. Theor. Comput. Sci. 329(1-3), 303–313 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    del Val, A.: On 2-SAT and renamable horn. In: AAAI: 17th National Conference on Artificial Intelligence, AAAI / MIT Press (2000)Google Scholar
  4. 4.
    Fernau, H.: A top-down approach to search-trees: Improved algorithmics for 3-hitting set. Electronic Colloquium on Computational Complexity, TR04-073 (2004)Google Scholar
  5. 5.
    Franco, J., Swaminathan, R.: On good algorithms for determining unsatisfiability of propositional formulas. Discrete Appl. Math. 130(2), 129–138 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Interian, Y.: Backdoor sets for random 3-sat. In: SAT (2003)Google Scholar
  7. 7.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Universität Tübingen (October 2002)Google Scholar
  8. 8.
    Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to horn and binary clauses. In: SAT (2004)Google Scholar
  9. 9.
    Nishimura, N., Ragde, P., Szeider, S.: Solving #SAT using vertex covers. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 396–409. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Porschen, S., Speckenmeyer, E.: Worst Case Bounds for Some NP-Complete Modified Horn-SAT Problems. In: H. Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 251–262. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Porschen, S., Speckenmeyer, E.: Satisfiability of mixed Horn formulas. Discrete Applied Mathematics 155(11), 1408–1419 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Ruan, Y., Kautz, H.A., Horvitz, E.: The backdoor key: A path to understanding problem hardness. In: AAAI, pp. 124–130 (2004)Google Scholar
  13. 13.
    Schöning, U.: A probabilistic algorithm for k-sat and constraint satisfaction problems. In: Symposium on Foundations of Computer Science (1999)Google Scholar
  14. 14.
    Szeider, S.: Matched Formulas and Backdoor Sets. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 94–99. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Wahlström, M.: Algorithms, measures, and upper bounds for satisfiability and related problems. PhD thesis, Linköping University, Dissertation no 1079 (2007)Google Scholar
  16. 16.
    Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: IJCAI (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Stephan Kottler
    • 1
  • Michael Kaufmann
    • 1
  • Carsten Sinz
    • 1
  1. 1.Wilhelm–Schickard–InstituteEberhard Karls Universität TübingenTübingenGermany

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