Skip to main content
SpringerLink
Log in
SpringerLink
Log in

The Parameterized Complexity of k-Flip Local Search for SAT and MAX SAT

  • Conference paper
Theory and Applications of Satisfiability Testing - SAT 2009 (SAT 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5584))

Abstract

Sat and Max Sat are among the most prominent problems for which local search algorithms have been successfully applied. A fundamental task for such an algorithm is to increase the number of clauses satisfied by a given truth assignment by flipping the truth values of at most k variables (k-flip local search). For a total number of n variables the size of the search space is of order n k and grows quickly in k; hence most practical algorithms use 1-flip local search only. In this paper we investigate the worst-case complexity of k-flip local search, considering k as a parameter: is it possible to search significantly faster than the trivial n k bound? In addition to the unbounded case we consider instances with a bounded number of literals per clause or where each variable occurs in a bounded number of clauses. We also consider the related problem that asks whether we can satisfy all clauses by flipping the truth values of at most k variables.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Dantsin, E., Goerdt, A., Hirsch, E.A., Kannan, R., Kleinberg, J.M., Papadimitriou, C.H., Raghavan, P., Schöning, U.: A deterministic (2 − 2/(k + 1))n algorithm for k-SAT based on local search. Theoret. Comput. Sci. 289(1), 69–83 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  3. Fellows, M.R.: Blow-Ups, Win/Win’s, and Crown Rules: Some New Directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  4. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  5. Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. J. ACM 48(6), 1184–1206 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hoos, H.H., Stützle, T.: Stochastic Local Search: Foundations and Applications. Elsevier/Morgan Kaufmann (2004)

    Google Scholar 

  7. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. of Computer and System Sciences 63(4), 512–530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? J. of Computer and System Sciences 37(1), 79–100 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khuller, S., Bhatia, R., Pless, R.: On local search and placement of meters in networks. SIAM J. Comput. 32(2), 470–487 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Krokhin, A.A., Marx, D.: On the hardness of losing weight. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 662–673. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  11. Marx, D.: Searching the k-change neighborhood for TSP is W[1]-hard. Oper. Res. Lett. 36(1), 31–36 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  13. Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. of Computer and System Sciences 67(4), 757–771 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Samer, M., Szeider, S.: Fixed-parameter tractability. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, ch. 13, pp. 425–454. IOS Press, Amsterdam (2009)

    Google Scholar 

  15. Yagiura, M., Ibaraki, T.: Analyses on the 2 and 3-flip neighborhoods for the MAX SAT. J. Comb. Optim. 3(1), 95–114 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Durham University, Durham, DH1 3LE, England, UK

    Stefan Szeider

Authors
  1. Stefan Szeider
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Science Department, Swansea University, Faraday Building, Singleton Park, SA2 8PP, Swansea, UK

    Oliver Kullmann

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Szeider, S. (2009). The Parameterized Complexity of k-Flip Local Search for SAT and MAX SAT. In: Kullmann, O. (eds) Theory and Applications of Satisfiability Testing - SAT 2009. SAT 2009. Lecture Notes in Computer Science, vol 5584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02777-2_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02777-2_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02776-5

  • Online ISBN: 978-3-642-02777-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation