Parameterized and Subexponential-Time Complexity of Satisfiability Problems and Applications

  • Iyad KanjEmail author
  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)


We study the parameterized and the subexponential-time complexity of the weighted and unweighted satisfiability problems on bounded-depth Boolean circuits. We establish relations between the subexponential-time complexity of the weighted and unweighted satisfiability problems, and use them to derive relations among the subexponential-time complexity of several \(\text {NP}\)-hard problem. For instance, we show that the weighted monotone satisfiability problem is solvable in subexponential time if and only if CNF-Sat is. The aforementioned result implies, via standard reductions, that several \(\text {NP}\)-hard problems are solvable in subexponential time if and only if CNF-Sat is. We also obtain threshold functions on structural circuit parameters including depth, number of gates, and fan-in, that lead to tight characterizations of the parameterized and the subexponential-time complexity of the circuit problems under consideration. For instance, we show that the weighted satisfiability problem is \(\text {FPT}\) on bounded-depth circuits with \(O(\log {n})\) gates, where \(n\) is the number of variables in the circuit, and is not \(\text {FPT}\) on bounded-depth circuits of \(\omega (\log {n})\) gates unless the Exponential Time Hypothesis (ETH) fails.


Subexponential Time Complexity Weighted Satisfiability Problem Exponential Time Hypothesis (ETH) Bounded Depth Circuits Tight Characterization 
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  1. 1.
    Alber, J., Bodlaender, H., Ferneau, H., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Brüggemann, T., Kern, W.: An improved deterministic local search algorithm for 3SAT. Theoret. Comput. Sci. 329, 303–313 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67(4), 789–807 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Calabro, C., Impagliazzo, R., Paturi, R.: A duality between clause width and clause density for SAT. In: IEEE Conference on Computational Complexity, pp. 252–260 (2006)Google Scholar
  5. 5.
    Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput. 201(2), 216–231 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, J., Huang, X., Kanj, I., Xia, G.: Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci. 72(8), 1346–1367 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, J., Kanj, I.A.: Parameterized complexity and subexponential-time computability. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds.) Fellows Festschrift 2012. LNCS, vol. 7370, pp. 162–195. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  8. 8.
    Chen, J., Kanj, I., Xia, G.: On parameterized exponential time complexity. Theoret. Comput. Sci. 410(27–29), 2641–2648 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chen, Y., Flum, J.: On miniaturized problems in parameterized complexity theory. Theoret. Comput. Sci. 351(3), 314–336 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chen, Y., Flum, J.: Subexponential time and fixed-parameter tractability: exploiting the miniaturization mapping. J. Logic Comput. 19(1), 89–122 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Chen, Y., Grohe, M.: An isomorphism between subexponential and parameterized complexity theory. SIAM J. Comput. 37(4), 1228–1258 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlström, M.: On problems as hard as CNF-SAT. In: IEEE Conference on Computational Complexity, pp. 74–84. IEEE (2012)Google Scholar
  13. 13.
    Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fernau, H.: edge dominating set: efficient enumeration-based exact algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 142–153. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized complexity theory. In: Brauer, W., Rozenberg, G., Salomaa, A. (eds.) Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)Google Scholar
  16. 16.
    Flüm, J., Grohe, M.: Parameterized Complexity Theory. Springer-verlag, Berlin (2010)Google Scholar
  17. 17.
    Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 760–776 (2011)Google Scholar
  20. 20.
    Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Patrascu, M., Williams, R.: On the possibility of faster SAT algorithms. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1065–1075 (2010)Google Scholar
  22. 22.
    Szeider, S.: The parameterized complexity of \(k\)-flip local search for SAT and MAX SAT. Discrete Optim. 8(1), 139–145 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.School of ComputingDePaul UniversityChicagoUSA
  2. 2.Vienna University of TechnologyViennaAustria

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