Advertisement

A New Bound for 3-Satisfiable Maxsat and Its Algorithmic Application

  • Gregory Gutin
  • Mark Jones
  • Anders Yeo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6914)

Abstract

Let F be a CNF formula with n variables and m clauses. F is t-satisfiable if for any t clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least \(\frac{2}{3}\) of its clauses can be satisfied by a truth assignment. Yannakakis’s proof utilizes the fact that \(\frac{2}{3}m\) is a lower bound on the expected number of clauses satisfied by a random truth assignment over a certain distribution. A CNF formula F is called expanding if for every subset X of the variables of F, the number of clauses containing variables of X is not smaller than |X|. In this paper we strengthen the \(\frac{2}{3}m\) bound for expanding 3-satisfiable CNF formulas by showing that for every such formula F at least \(\frac{2}{3}m + \rho n\) clauses of F can be satisfied by a truth assignment, where ρ( > 0.0019) is a constant. Our proof uses a probabilistic method with a sophisticated distribution for truth values. We use the bound \(\frac{2}{3}m + \rho n\) and results on matching autarkies to obtain a new lower bound on the maximum number of clauses that can be satisfied by a truth assignment in any 3-satisfiable CNF formula.

We use our results above to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MaxSat-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least \(\frac{2}{3}m + k\) clauses, where k is the parameter. Note that Mahajan and Raman (1999) asked whether 2-S-MaxSat-AE, the corresponding problem for 2-satisfiable formulas, is fixed-parameter tractable. Crowston and the authors of this paper proved in [9] that 2-S-MaxSat-AE is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. 2-S-MaxSat-AE appears to be easier than 3-S-MaxSat-AE and, unlike this paper, [9] uses only deterministic combinatorial arguments.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N., Fernau, H.: Kernels: Annotated, proper and induced. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 264–275. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Comb. Theory, Ser. A 43(2), 196–204 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-k-SAT Above a Tight Lower Bound. Algorithmica, (in press), doi:10.1007/s00453-010-9428-7; A preliminary version in Proc. SODA 2010, pp. 511–517Google Scholar
  4. 4.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley, Chichester (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Büning, H.K., Kullmann, O.: Minimal Unsatisfiability and Autarkies. In: Handbook of Satisfiability, ch. 11, pp. 339–401., doi:10.3233/978-1-58603-929-5-339Google Scholar
  6. 6.
    Chen, Y., Flum, J., Müller, M.: Lower bounds for kernelizations and other preprocessing procedures. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 118–128. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Crowston, R., Gutin, G., Jones, M.: Note on Max Lin-2 above Average. Inform. Proc. Lett. 110, 451–454 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crowston, R., Gutin, G., Jones, M., Kim, E.J., Ruzsa, I.Z.: Systems of linear equations over \(\mathbb{F}_2\) and problems parameterized above average. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 164–175. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Crowston, R., Gutin, G., Jones, M., Yeo, A.: A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and Its Algorithmic Application. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 84–94. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoret. Comput. Sci. 289(1), 503–516 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  13. 13.
    Gutin, G., Jones, M., Yeo, A.: A new bound for 3-Satisfiable MaxSat and its algorithmic application, arXiv:1104.2818 (April 2011)Google Scholar
  14. 14.
    Huang, M.A., Lieberherr, K.J.: Implications of forbidden structures for extremal algorithmic problems. Theoret. Comput. Sci. 40, 195–210 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kim, E.J., Williams, R.: Improved Parameterized Algorithms for Constraint Satisfaction, arXiv:1008.0213 (August 2010)Google Scholar
  16. 16.
    Král, D.: Locally satisfiable formulas. In: Proc. SODA 2004, pp. 330–339 (2004)Google Scholar
  17. 17.
    Kullmann, O.: Lean clause-sets: Generalizations of minimally unsatisfiable clause-sets. Discrete Applied Mathematics 130, 209–249 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lieberherr, K.J., Specker, E.: Complexity of partial satisfaction. J. ACM 28(2), 411–421 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lieberherr, K.J., Specker, E.: Complexity of partial satisfaction, II. Tech. Report 293 of Dept. of EECS, Princeton Univ. (1982)Google Scholar
  20. 20.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. Syst. Sci. 75(2), 137–153 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Monien, B., Speckenmeyer, E.: Solving satisfiability in less than 2n steps. Discr. Appl. Math. 10, 287–295 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  24. 24.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. J. Comput. Syst. Sci. 69(4), 656–674 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Trevisan, L.: On local versus global satisfiability. SIAM J. Discret. Math. 17(4), 541–547 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yannakakis, M.: On the approximation of maximum satisfiability. J. Algorithms 17, 475–502 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Anders Yeo
    • 1
  1. 1.Royal Holloway, University of LondonUnited Kingdom

Personalised recommendations