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On the Parameterized Complexity of Exact Satisfiability Problems

  • Joachim Kneis
  • Daniel Mölle
  • Stefan Richter
  • Peter Rossmanith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

For many problems, the investigation of their parameterized complexity provides an interesting and useful point of view. The most obvious natural parameterization for the maximum satisfiability problem—the number of satisfiable clauses—makes little sense, because at least half of the clauses can be satisfied in any formula. We look at two optimization variants of the exact satisfiability problem, where a clause is only said to be fulfilled iff exactly one of its literals is set to true. Interestingly, these variants behave quite differently. In the case of ResMaxExactSAT, where over-satisfied clauses are entirely forbidden, we show fixed parameter tractability. On the other hand, if we choose to ignore over-satisfied clauses, the MaxExactSAT problem is obtained. Surprisingly, it is W[1]-complete. Still, restricted variants of the problem turn out to be tractable.

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References

  1. 1.
    Byskov, J.M., Madsen, B.A., Skjernaa, B.: New algorithms for exact satisfiability. Technical Report RS-03-30, BRICS (October 2003)Google Scholar
  2. 2.
    Dahllöf, V., Johnson, P., Beigel, R.: Algorithms for four variants of the exact satisfiability problem. Theoretical Comp. Sci. 320(2-3), 373–394 (2004)zbMATHCrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  4. 4.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified np-complete graph problems. Theoretical Computer Science 1, 237–267 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gramm, J., Hirsch, E.A., Niedermeier, R., Rossmanith, P.: New worst-case upper bounds for MAX-2-SAT with application to MAX-CUT. Discrete Applied Mathematics 130(2), 139–155 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. Journal of Algorithms 50(1), 106–117 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Communications, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  8. 8.
    Kulikov, A.S.: An upper bound O(20.16254n): A simpler proof. Zapiski nauchnyh seminarov POMI 293, 118–128 (2002)Google Scholar
  9. 9.
    Madsen, B.: An algorithm for exact satisfiability analysed with the number of clauses as parameter. Technical Report RS-04-18, BRICS (2004)Google Scholar
  10. 10.
    Madsen, B.A., Rossmanith, P.: Maximum exact satisfiability: NP-completeness proofs and exact algorithms. Technical Report RS-04-19, BRICS (October 2004)Google Scholar
  11. 11.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms 31, 335–354 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  13. 13.
    Porschen, S., Randerath, B., Speckenmeyer, E.: Exact 3-satisfiability is decidable in time O(20.16254n). In: Proc. 5th Int. Symp. on the Theory and Appl. Satisfiabilty testing (SAT2002), pp. 231–235 (2002)Google Scholar
  14. 14.
    Schnorr, C.P.: Satisfiability is quasilinear complete in NQL. J. ACM 25, 136–145 (1978)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Joachim Kneis
    • 1
  • Daniel Mölle
    • 1
  • Stefan Richter
    • 1
  • Peter Rossmanith
    • 1
  1. 1.Dept. of Computer ScienceRWTH Aachen UniversityGermany

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