A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and Its Algorithmic Application

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Anders Yeo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6478)


For a formula F in conjunctive normal form (CNF), let sat(F) be the maximum number of clauses of F that can be satisfied by a truth assignment, and let m be the number of clauses in F. It is well-known that for every CNF formula F, sat(F) ≥ m/2 and the bound is tight when F consists of conflicting unit clauses (x) and \((\bar{x})\). Since each truth assignment satisfies exactly one clause in each pair of conflicting unit clauses, it is natural to reduce F to the unit-conflict free (UCF) form. If F is UCF, then Lieberherr and Specker (J. ACM 28(2):411-421, 1981) proved that \({\rm sat}(F)\ge {\hat{\phi}} m\), where \({\hat{\phi}} =(\sqrt{5}-1)/2\).

We introduce another reduction that transforms a UCF CNF formula F into a UCF CNF formula F′, which has a complete matching, i.e., there is an injective map from the variables to the clauses, such that each variable maps to a clause containing that variable or its negation. The formula F′ is obtained from F by deleting some clauses and the variables contained only in the deleted clauses. We prove that \({\rm sat}(F) \ge {\hat{\phi}} m + (1-{\hat{\phi}})(m-m') + n'(2-3{\hat{\phi}})/2\), where n′ and m′ are the number of variables and clauses in F′, respectively. This improves the Lieberherr-Specker lower bound on sat(F).

We show that our new bound has an algorithmic application by considering the following parameterized problem: given a UCF CNF formula F decide whether sat(F) \(\ge {\hat{\phi}} m + k,\) where k is the parameter. This problem was introduced by Mahajan and Raman (J. Algorithms 31(2):335–354, 1999) who conjectured that the problem is fixed-parameter tractable, i.e., it can be solved in time f(k)(nm) O(1) for some computable function f of k only. We use the new bound to show that the problem is indeed fixed-parameter tractable by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most \(\lfloor (7+3\sqrt{5})k\rfloor\) variables.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.
    Chen, Y., Flum, J., Müller, M.: Lower bounds for kernelizations and other preprocessing procedures. In: Proc. CiE 2009, vol. 5635, pp. 118–128 (2009)Google Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  5. 5.
    Huang, M.A., Lieberherr, K.J.: Implications of forbidden structures for extremal algorithmic problems. Theoret. Comput. Sci. 40, 195–210 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Käppeli, C., Scheder, D.: Partial satisfaction of k-satisfiable formulas. Electronic Notes in Discrete Math. 29, 497–501 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Král, D.: Locally satisfiable formulas. In: Proc. SODA 2004, pp. 330–339 (2004)Google Scholar
  8. 8.
    Lieberherr, K.J., Specker, E.: Complexity of partial satisfaction. J. ACM 28(2), 411–421 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lieberherr, K.J., Specker, E.: Complexity of partial satisfaction, II. Tech. Report 293 of Dept. of EECS, Princeton Univ. (1982)Google Scholar
  10. 10.
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31(2), 335–354 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  12. 12.
    Trevisan, L.: On local versus global satisfiability. SIAM J. Discret. Math. 17(4), 541–547 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yannakakis, M.: On the approximation of maximum satisfiability. J. Algorithms 17, 475–502 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, Englewood Cliffs (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Robert Crowston
    • 1
  • Gregory Gutin
    • 1
  • Mark Jones
    • 1
  • Anders Yeo
    • 1
  1. 1.Royal HollowayUniversity of LondonUnited Kingdom

Personalised recommendations