Upper and Lower Bounds for Weak Backdoor Set Detection

  • Neeldhara Misra
  • Sebastian Ordyniak
  • Venkatesh Raman
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7962)


We obtain upper and lower bounds for running times of exponential time algorithms for the detection of weak backdoor sets of 3CNF formulas, considering various base classes. These results include (omitting polynomial factors), (i) a 4.54 k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Horn formulas; (ii) a 2.27 k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Krom formulas. These bounds improve an earlier known bound of 6 k . We also prove a 2 k lower bound for these problems, subject to the Strong Exponential Time Hypothesis.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boros, E., Hammer, P.L., Sun, X.: Recognition of q-Horn formulae in linear time. Discr. Appl. Math. 55(1), 1–13 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Crama, Y., Ekin, O., Hammer, P.L.: Variable and term removal from Boolean formulae. Discr. Appl. Math. 75(3), 217–230 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)CrossRefGoogle Scholar
  4. 4.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)Google Scholar
  5. 5.
    Franco, J., Van Gelder, A.: A perspective on certain polynomial time solvable classes of satisfiability. Discr. Appl. Math. 125, 177–214 (2003)zbMATHCrossRefGoogle Scholar
  6. 6.
    Gaspers, S., Ordyniak, S., Ramanujan, M.S., Saurabh, S., Szeider, S.: Backdoors to q-horn. In: Portier, N., Wilke, T. (eds.) Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science (STACS). Leibniz International Proceedings in Informatics (LIPIcs), vol. 20, pp. 67–79 (2013)Google Scholar
  7. 7.
    Gaspers, S., Szeider, S.: Backdoors to acyclic SAT. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 363–374. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Gaspers, S., Szeider, S.: Backdoors to satisfaction. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift 2012. LNCS, vol. 7370, pp. 287–317. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. of Computer and System Sciences 62(2), 367–375 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. of Computer and System Sciences 63(4), 512–530 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bulletin of the European Association for Theoretical Computer Science 105, 41–72 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its. Applications. Oxford University Press, Oxford (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    Niedermeier, R., Rossmanith, P.: An efficient fixed-parameter algorithm for 3-hitting set. J. Discrete Algorithms 1(1), 89–102 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Proceedings of SAT 2004 (Seventh International Conference on Theory and Applications of Satisfiability Testing), Vancouver, BC, Canada, May 10-13, pp. 96–103 (2004)Google Scholar
  15. 15.
    Raman, V., Shankar, B.S.: Improved fixed-parameter algorithm for the minimum weight 3-SAT problem. In: Ghosh, S.K., Tokuyama, T. (eds.) WALCOM 2013. LNCS, vol. 7748, pp. 265–273. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  16. 16.
    Williams, R., Gomes, C., Selman, B.: Backdoors to typical case complexity. In: Gottlob, G., Walsh, T. (eds.) Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, IJCAI 2003, pp. 1173–1178. Morgan Kaufmann (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Neeldhara Misra
    • 1
  • Sebastian Ordyniak
    • 2
  • Venkatesh Raman
    • 3
  • Stefan Szeider
    • 4
  1. 1.Indian Institute of ScienceBangaloreIndia
  2. 2.Masaryk University BrnoCzech Republic
  3. 3.Institute of Mathematical SciencesChennaiIndia
  4. 4.Vienna University of TechnologyAustria

Personalised recommendations