Skip to main content
SpringerLink
Log in

Parameterized and Subexponential-Time Complexity of Satisfiability Problems and Applications

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8881))

  • 1429 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For \(k\)-CNF-Sat the instance size is polynomial in \(n\), and hence, gets absorbed in the exponential term \(O(2^{c_kn})\). Throughout the paper, we will omit the polynomial factor in the instance size whenever it is polynomial in \(n\).

  2. 2.

    A padding argument is a general tool that is used in complexity theory to extend a result to a larger class of problems. For our purpose in this paper, the padding argument works by adding/padding a “dummy” part to the instance to create an equivalent new instance in which a relation holds true between certain parameters in the new instance. We will use the padding argument a couple of times in this paper, and skip the details when it is clear how the instance can be padded.

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brüggemann, T., Kern, W.: An improved deterministic local search algorithm for 3SAT. Theoret. Comput. Sci. 329, 303–313 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. J. Comput. Syst. Sci. 67(4), 789–807 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  8. Chen, J., Kanj, I., Xia, G.: On parameterized exponential time complexity. Theoret. Comput. Sci. 410(27–29), 2641–2648 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, Y., Flum, J.: On miniaturized problems in parameterized complexity theory. Theoret. Comput. Sci. 351(3), 314–336 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Y., Flum, J.: Subexponential time and fixed-parameter tractability: exploiting the miniaturization mapping. J. Logic Comput. 19(1), 89–122 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, Y., Grohe, M.: An isomorphism between subexponential and parameterized complexity theory. SIAM J. Comput. 37(4), 1228–1258 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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. Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer, New York (2013)

    Book  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Flüm, J., Grohe, M.: Parameterized Complexity Theory. Springer-verlag, Berlin (2010)

    Google Scholar 

  17. Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. Papadimitriou, C., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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. Szeider, S.: The parameterized complexity of \(k\)-flip local search for SAT and MAX SAT. Discrete Optim. 8(1), 139–145 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing, DePaul University, Chicago, USA

    Iyad Kanj

  2. Vienna University of Technology, Vienna, Austria

    Stefan Szeider

Authors
  1. Iyad Kanj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefan Szeider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iyad Kanj .

Editor information

Editors and Affiliations

  1. Zhejiang Normal University, Jinhua, Zhejiang, China

    Zhao Zhang

  2. University of Texas, Tyler, Texas, USA

    Lidong Wu

  3. University of Texas, Dallas, Texas, USA

    Wen Xu

  4. University of Texas, Dallas, USA

    Ding-Zhu Du

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Kanj, I., Szeider, S. (2014). Parameterized and Subexponential-Time Complexity of Satisfiability Problems and Applications. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_48

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-12691-3_48

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12690-6

  • Online ISBN: 978-3-319-12691-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation