Skip to main content
SpringerLink
Log in

Efficient Recognition of Random Unsatisfiable k-SAT Instances by Spectral Methods

  • Conference paper
  • First Online:
STACS 2001 (STACS 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2010))

Included in the following conference series:

Abstract

It is known that random k-SAT instances with at least cn clauses where c = c k is a suitable constant are unsatisfiable (with high probability). We consider the problem to certify efficiently the unsatis- fiability of such formulas. A result of Beame et al. shows that k-SAT instances with at least n k-1= log n clauses can be certified unsatisfiable in polynomial time. We employ spectral methods to improve on this: We present a polynomial time algorithm which certifies random k-SAT instances for k even with at least 2k . (k/2)7 . (ln n)7 . n k/2 = n (k/2)+o(1) clauses as unsatisfiable (with high probability).

Partially supported by a USA-Israeli BSF grant.

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. Dimitris Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings SToC 2000, ACM.

    Google Scholar 

  2. Dimitris Achlioptas, Ehud Friedgut. A threshold for random k-colourability. Random Structures and Algorithms 1999.

    Google Scholar 

  3. Dimitris Achlioptas, Mike Molloy. Analysis of a list colouring algorithm on a random graph. In Proceedings FoCS 1997, IEEE.

    Google Scholar 

  4. Dimitris Achlioptas, Mike Molloy. Almost all graphs with 2.522n edges are not 3-colourable. Undated manuscript.

    Google Scholar 

  5. Noga Alon, Nabil Kahale. A spectral technique for colouring random 3-colourable graphs (preliminary version). In Proceedings 26th SToC,1994, ACM, 346–355.

    Google Scholar 

  6. Noga Alon, Joel H. Spencer. The Probabilistic Method. Wiley & Sons Inc., 1992.

    Google Scholar 

  7. Paul Beame, Richard Karp, Toniann Pitassi, Michael Saks. On the complexity of unsatisfiability proofs for random k-CNF formulas. In Proceedings 30th SToC,1998, ACM, 561–571.

    Google Scholar 

  8. Paul Beame, Toniann Pitassi. Simplified and improved resolution lower bounds. In Proceedings 37th FoCS,1996, IEEE, 274–282.

    Google Scholar 

  9. Bela Bollobas. Random Graphs. Academic Press, 1985.

    Google Scholar 

  10. Manuel Blum, Richard Karp, Oliver Vornberger, Christos H. Papadimitriou, Mihalis Yannakakis. The complexity of testing whether a graph is a superconcentrator. Information Processing Letters 13, 1981, 164–167.

    Article  MATH  MathSciNet  Google Scholar 

  11. Fan R. K. Chung. Spectral Graph Theory. American Mathematical Society, 1997.

    Google Scholar 

  12. Vasek Chvatal, Bruce Reed. Mick gets some (the odds are on his side). In Proceedings 33nd FoCS,1992, IEEE, 620–627.

    Google Scholar 

  13. Vasek Chvatal, Endre Szemeredi. Many hard examples for resolution. Journal of the ACM 35(4), 1988, 759–768.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. M. Crawford, L. D. Auton. Experimental results on the crossover point in random 3-SAT. Artificial Intelligence 81, 1996.

    Google Scholar 

  15. Olivier Dubois, Yacine Boufkhad, Jacques Mandler. Typical random 3-SAT formulae and the satisfiability threshold. In Proceedings SoDA 2000, SIAM.

    Google Scholar 

  16. Paul E. Dunne, Michele Zito. An improved upper bound for the non-3-colourability threshold. Information Processing Letters 1998.

    Google Scholar 

  17. Xudong Fu. The complexity of the resolution proofs for the random set of clauses. Computational Complexity 1998.

    Google Scholar 

  18. Ehud Friedgut. Necessary and suficient conditions for sharp thresholds of graph properties and the k-SAT problem. Journal of the American Mathematical Society 12, 1999, 1017–1054.

    Article  MATH  MathSciNet  Google Scholar 

  19. Alan M. Frieze, Stephen Suen. Analysis of two simple heuristics on a random instance of k-SAT. Journal of Algorithms 20(2), 1996, 312–355.

    Article  MATH  MathSciNet  Google Scholar 

  20. Z. Furedi, J. Komlos. The eigenvalues of random symmetric matrices. Combinatorica 1(3), 1981, 233–241.

    Article  MathSciNet  Google Scholar 

  21. Andreas Goerdt. A threshold for unsatisfiability. Journal of Computer and System Sciences 53, 1996, 469–486.

    Article  MATH  MathSciNet  Google Scholar 

  22. Russel Impagliazzo, Moni Naor. Efficient cryptographic schemes provably as secure as subset sum. Journal of Cryptology 9, 1996, 199–216.

    Article  MATH  MathSciNet  Google Scholar 

  23. Ferenc Juhasz. The asymptotic behaviour of Lovasz theta function for random graphs. Combinatorica 2(2), 1982, 153–155.

    Article  MATH  MathSciNet  Google Scholar 

  24. Michael Krivelevich, Van H. Vu. Approximating the independence number and the chromatic number in expected polynomial time. In Proceedings ICALP 2000, LNCS 1853, 13–24.

    Google Scholar 

  25. Lefteris M. Kirousis, Evangelos Kranakis, Danny Krizanc, Yiannis Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms 12(3), 1998, 253–269.

    Article  MATH  MathSciNet  Google Scholar 

  26. A. D. Petford, Dominic Welsh. A Randomised 3-colouring algorithm. Discrete Mathematics 74, 1989, 253–261.

    Article  MathSciNet  MATH  Google Scholar 

  27. Bart Selman, David G. Mitchell, Hector J. Levesque. Generating hard satisfiability problems. Artificial Intelligence 81(1-2), 1996, 17–29.

    Article  MathSciNet  Google Scholar 

  28. Gilbert Strang. Linear Algebra and its Applications. Harcourt Brace Jovanovich, Publishers, San Diego, 1988.

    Google Scholar 

  29. J. H. van Lint, R. M. Wilson. A Course in Combinatorics. Cambridge University Press, 1992.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Andreas Goerdt

  2. Fakultät für Informatik, TU Chemnitz, 09107, Chemnitz, Germany

    Michael Krivelevich

Authors
  1. Andreas Goerdt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Krivelevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. I3S & INRIA Sophia Antipolis INRIA, CNRS, 2004 Route des Lucioles, 06902, Sophia-Antipolis, France

    Afonso Ferreira

  2. Institut für Theoretische Informatik, Fakultät Informatik, TU Dresden, 01062, Dresden, Germany

    Horst Reichel

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Goerdt, A., Krivelevich, M. (2001). Efficient Recognition of Random Unsatisfiable k-SAT Instances by Spectral Methods. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_26

Download citation

  • DOI: https://doi.org/10.1007/3-540-44693-1_26

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41695-1

  • Online ISBN: 978-3-540-44693-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation