Solving minones-2-sat as Fast as vertex cover

  • Neeldhara Misra
  • N. S. Narayanaswamy
  • Venkatesh Raman
  • Bal Sri Shankar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)


The problem of finding a satisfying assignment for a 2-SAT formula that minimizes the number of variables that are set to 1 (min ones 2–sat) is NP-complete. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k.

In this paper, we present a polynomial-time reduction from min ones 2–sat to vertex cover without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k variables kernel subsuming these results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover.




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AK07]
    Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. In: Dehne, F.K.H.A., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 434–445. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. [BJG08]
    Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, algorithms and applications. Springer Publishing Company, Heidelberg (2008) (incorporated)Google Scholar
  3. [CKX06]
    Chen, J., Kanj, I.A., Xia, G.: Improved parameterized upper bounds for vertex cover. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 238–249. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. [CLRS01]
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms. MIT Press, Cambridge (2001)MATHGoogle Scholar
  5. [DF99]
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (November 1999)Google Scholar
  6. [DJ02]
    Dahllöf, V., Jonsson, P.: An algorithm for counting maximum weighted independent sets and its applications. In: SODA, pp. 292–298 (2002)Google Scholar
  7. [GP92]
    Gusfield, D., Pitt, L.: A bounded approximation for the minimum cost 2-sat problem. Algorithmica 8, 103–117 (1992)MATHCrossRefMathSciNetGoogle Scholar
  8. [HMNT93]
    Hochbaum, D., Meggido, N., Naor, J., Tamir, A.: Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality. Mathematical Programming 62, 69–83 (1993)CrossRefMathSciNetGoogle Scholar
  9. [Hoc97]
    Hochbaum, D.S. (ed.): Approximation algorithms for NP-hard problems. PWS Publishing Co., Boston (1997)Google Scholar
  10. [KLR09]
    Kneis, J., Langer, A., Rossmanith, P.: A fine-grained analysis of a simple independent set algorithm. In: Kannan, R., Kumar, N. (eds.) IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2009), Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 4, pp. 287–298. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2009)Google Scholar
  11. [KW09]
    Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 264–275. Springer, Heidelberg (2009)Google Scholar
  12. [KWar]
    Kratsch, S., Wahlström, M.: Preprocessing of min ones problems: A dichotomy. In: 37th International Colloquium on Automata, Languages and Programming, ICALP (to appear 2010)Google Scholar
  13. [MR99]
    Mahajan, M., Raman, V.: Parameterizing above guaranteed values: Maxsat and maxcut. J. Algorithms 31(2), 335–354 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. [Nie06]
    Niedermeier, R.: Invitation to fixed parameter algorithms (oxford lecture series in mathematics and its applications). Oxford University Press, USA (March 2006)CrossRefGoogle Scholar
  15. [NR03]
    Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms 47(2), 63–77 (2003)MATHMathSciNetGoogle Scholar
  16. [NT75]
    Nemhauser, G.L., Trotter, L.E.: Vertex packings: Structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Neeldhara Misra
    • 1
  • N. S. Narayanaswamy
    • 1
  • Venkatesh Raman
    • 1
  • Bal Sri Shankar
    • 1
  1. 1.Institute of Mathematical SciencesChennaiIndia

Personalised recommendations