Advertisement

Parameterized Complexity of MaxSat above Average

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Venkatesh Raman
  • Saket Saurabh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

In MaxSat, we are given a CNF formula F with n variables and m clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let r 1,…, r m be the number of literals in the clauses of F. Then \({\rm asat}(F)=\sum_{i=1}^m (1-2^{-r_i})\) is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independently). It is well-known that, in polynomial time, one can find a truth assignment satisfying at least asat(F) clauses. In the parameterized problem MaxSat-AA, we are to decide whether there is a truth assignment satisfying at least asat(F) + k clauses, where k is the (nonnegative) parameter. We prove that MaxSat-AA is para-NP-complete and thus, MaxSat-AA is not fixed-parameter tractable unless P=NP. This is in sharp contrast to the similar problem MaxLin2-AA which was recently proved to be fixed-parameter tractable by Crowston et al. (FSTTCS 2011).

In fact, we consider a more refined version of MaxSat-AA, Max- r(n)-Sat-AA, where r j  ≤ r(n) for each j. Alon et al. (SODA 2010) proved that if r = r(n) is a constant, then Max- r -Sat-AA is fixed-parameter tractable. We prove that Max- r(n)-Sat-AA is para-NP-complete for r(n) = ⌈logn⌉. We also prove that assuming the exponential time hypothesis, Max- r(n)-Sat-AA is not fixed-parameter tractable already for any r(n) ≥ loglogn + φ(n), where φ(n) is any unbounded strictly increasing function. This lower bound on r(n) cannot be decreased much further as we prove that Max- r(n)-Sat-AA is fixed-parameter tractable for any r(n) ≤ loglogn − logloglogn − φ(n), where φ(n) is any unbounded strictly increasing function. The proof uses some results on MaxLin2-AA.

Keywords

Parameterized Complexity Constraint Satisfaction Problem Vertex Cover Truth Assignment Reduction Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. Algorithmica 61, 638–655 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley (2000)Google Scholar
  3. 3.
    Charikar, M., Guruswami, V., Manokaran, R.: Every permutation CSP of arity 3 is approximation resistant. In: Proc. Computational Complexity 2009, pp. 62–73 (2009)Google Scholar
  4. 4.
    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 Applications. Algorithmica, doi:10.1007/s00453-011-9550-1Google Scholar
  5. 5.
    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
  6. 6.
    Crowston, R., Fellows, M., Gutin, G., Jones, M., Rosamond, F., Thomassé, S., Yeo, A.: Simultaneously Satisfying Linear Equations Over \(\mathbb{F}_2\): MaxLin2 and Max-r-Lin2 Parameterized Above Average. In: Proc. FSTTCS 2011. LIPICS, vol. 13, pp. 229–240 (2011)Google Scholar
  7. 7.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On Multiway Cut Parameterized above Lower Bounds. In: Rossmanith, P. (ed.) IPEC 2011. LNCS, vol. 7112, pp. 1–12. Springer, Heidelberg (2012)Google Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  10. 10.
    Guruswami, V., Håstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: Every ordering CSP is approximation resistant. SIAM J. Comput. 40(3), 878–914 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Guruswami, V., Manokaran, R., Raghavendra, P.: Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph. In: Proc. FOCS 2008, pp. 573–582 (2008)Google Scholar
  12. 12.
    Gutin, G., van Iersel, L., Mnich, M., Yeo, A.: Every ternary permutation constraint satisfaction problem parameterized above average has a kernel with a quadratic number of variables. J. Comput. System Sci. 78, 151–163 (2012)CrossRefGoogle Scholar
  13. 13.
    Gutin, G., Jones, M., Yeo, A.: A New Bound for 3-Satisfiable Maxsat and Its Algorithmic Application. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 138–147. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: A probabilistic approach to problems parameterized above or below tight bounds. J. Comput. Sys. Sci. 77, 422–429 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Sys. Sci. 62, 367–375 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Sys. Sci. 63, 512–530 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. STOC 2002, pp. 767–775 (2002)Google Scholar
  18. 18.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Mahajan, M., Raman, V., Sikdar, S.: Parameterizing above or below guaranteed values. J. Comput. Sys. Sci. 75(2), 137–153 (2009); In: Bodlaender, H.L., Langston, M.A. (eds.): IWPEC 2006. LNCS, vol. 4169, pp. 38–49. Springer, Heidelberg (2006)Google Scholar
  20. 20.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  21. 21.
    Raman, V., Ramanujan, M.S., Saurabh, S.: Paths, Flowers and Vertex Cover. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 382–393. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Tovey, C.A.: A simplified satisfiability problem. Discr. Appl. Math. 8, 85–89 (1984)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Robert Crowston
    • 1
  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Venkatesh Raman
    • 2
  • Saket Saurabh
    • 2
  1. 1.Royal Holloway, University of LondonEghamUK
  2. 2.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations