Fixed-Parameter Tractability of Satisfying beyond the Number of Variables

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Venkatesh Raman
  • Saket Saurabh
  • Anders Yeo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7317)


We consider a CNF formula F as a multiset of clauses: F = {c 1,…, c m }. The set of variables of F will be denoted by V(F). Let B F denote the bipartite graph with partite sets V(F) and F and an edge between v ∈ V(F) and c ∈ F if v ∈ c or \(\bar{v} \in c\). The matching number ν(F) of F is the size of a maximum matching in B F . In our main result, we prove that the following parameterization of MaxSat is fixed-parameter tractable: Given a formula F, decide whether we can satisfy at least ν(F) + k clauses in F, where k is the parameter.

A formula F is called variable-matched if ν(F) = |V(F)|. Let δ(F) = |F| − |V(F)| and δ *(F) =  max F′ ⊆ F δ(F′). Our main result implies fixed-parameter tractability of MaxSat parameterized by δ(F) for variable-matched formulas F; this complements related results of Kullmann (2000) and Szeider (2004) for MaxSat parameterized by δ *(F).

To prove our main result, we obtain an O((2e)2k k O(logk) (m + n) O(1))-time algorithm for the following parameterization of the Hitting Set problem: given a collection \(\cal C\) of m subsets of a ground set U of n elements, decide whether there is X ⊆ U such that C ∩ X ≠ ∅ for each \(C\in \cal C\) and |X| ≤ m − k, where k is the parameter. This improves an algorithm that follows from a kernelization result of Gutin, Jones and Yeo (2011).


Polynomial Time Bipartite Graph Polynomial Kernel Maximum Match Truth Assignment 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. System Sci. 75(8), 423–434 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel Bounds for Disjoint Cycles and Disjoint Paths. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 635–646. Springer, Heidelberg (2009)CrossRefGoogle 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.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through Colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  7. 7.
    Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theor. Comput. Sci. 289(1), 503–516 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fleischner, H., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Electronic Colloquium on Computational Complexity (ECCC) 7(49) (2000)Google Scholar
  9. 9.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer (2006)Google Scholar
  10. 10.
    Gutin, G., Jones, M., Yeo, A.: Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems. Theor. Comput. Sci. 412(41), 5744–5751 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kleine Büning, H.: On subclasses of minimal unsatisfiable formulas. Discrete Applied Mathematics 107(1-3), 83–98 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kleine Büning, H., Kullmann, O.: Minimal Unsatisfiability and Autarkies. In: Handbook of Satisfiability, ch. 11, pp. 339–401Google Scholar
  13. 13.
    Kullmann, O.: An application of matroid theory to the sat problem. In: IEEE Conference on Computational Complexity, pp. 116–124 (2000)Google Scholar
  14. 14.
    Kullmann, O.: Lean clause-sets: Generalizations of minimally unsatisfiable clause-sets. Discr. Appl. Math. 130, 209–249 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lovász, L., Plummer, M.D.: Matching theory. AMS Chelsea Publ. (2009)Google Scholar
  16. 16.
    Monien, B., Speckenmeyer, E.: Solving satisfiability in less than 2n steps. Discr. Appl. Math. 10, 287–295 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Papadimitriou, C.H., Wolfe, D.: The complexity of facets resolved. J. Comput. Syst. Sci. 37(1), 2–13 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Srinivasan, A.: Improved approximations of packing and covering problems. In: STOC 1995, pp. 268–276 (1995)Google Scholar
  19. 19.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. J. Comput. Syst. Sci. 69(4), 656–674 (2004)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
  • Anders Yeo
    • 3
  1. 1.Royal Holloway, University of LondonEghamUK
  2. 2.The Institute of Mathematical SciencesChennaiIndia
  3. 3.University of JohannesburgSouth Africa

Personalised recommendations