Advertisement

Satisfiability Thresholds beyond k −XORSAT

  • Andreas Goerdt
  • Lutz Falke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

We consider random systems of equations x 1 + … + x k  = a, 0 ≤ a ≤ 2 which are interpreted as equations modulo 3. We show for k ≥ 15 that the satisfiability threshold of such systems occurs where the 2 −core has density 1. We show a similar result for random uniquely extendible constraints over 4 elements. Our results extend previous results of Dubois/Mandler for equations mod 2 and k = 3 and Connamacher/Molloy for uniquely extendible constraints over a domain of 4 elements with k = 3 arguments.

The proof is based on variance calculations, using a technique introduced by Dubois/Mandler. However, several additional observations (of independent interest) are necessary.

Keywords

Threshold Density Laplace Method Local Limit Theorem Sharp Threshold Equation Modulo 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Achlioptas, D., Moore, C.: Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold. SIAM J. Comput. 36(3), 740–762 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Achlioptas, D., Ibrahimi, M., Kanoria, Y., Kraning, M., Molloy, M., Montanari, A.: The Set of Solutions of Random XORSAT Formulae. In: Proceedings SoDA 2012 (2012)Google Scholar
  3. 3.
    Bhattacharya, N., Ranga Rao, R.: Normal Approximation and Asymptotic Expansions. Robert E. Krieger Publishing Company (1986)Google Scholar
  4. 4.
    Braunstein, A., Mezard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability. arXiv:cs/0212002Google Scholar
  5. 5.
    de Bruijn, N.G.: Asymptotic Methods in Analysis. North Holland (1958)Google Scholar
  6. 6.
    Coja-Oghlan, A., Pachon-Pinzon, A.Y.: The Decimation Process in Random k-SAT. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 305–316. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Connamacher, H.: Exact thresholds for DPLL on random XOR-SAT and NP-complete extensions of XOR-SAT. Theoretical Computer Science (2011)Google Scholar
  8. 8.
    Connamacher, H., Molloy, M.: The exact satisfiability threshold for a potentially in tractable random constraint satisfaction problem. In: Proceedings 45th FoCS 2004, pp. 590–599 (2004)Google Scholar
  9. 9.
    Creignou, N., Daudé, H.: The SAT-UNSAT transition for random constraint satisfaction problems. Discrete Mathematics 309(8), 2085–2099Google Scholar
  10. 10.
    Diaz, J., et al.: On the satisfiability threshold of formulas with three literals per clause. Theoretical Computer Science 410, 2920–2934 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R., Rink, M.: Tight Thresholds for Cuckoo Hashing via XORSAT. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 213–225. Springer, Heidelberg (2010); arXiv:cs/0912.0287 (2009)CrossRefGoogle Scholar
  12. 12.
    Dubois, O., Mandler, J.: The 3 −XORSAT satisfiability threshold. In: Proceedings 43rd FoCS, p. 769 (2003)Google Scholar
  13. 13.
    Durrett, R.: Probability Theory: Theory and Examples. Wadsworth and Brooks (1991)Google Scholar
  14. 14.
    Friedgut, E.: Hunting for sharp thresholds. Random Struct. Algorithms 26(1-2), 37–51 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Goerdt, A.: On Random Betweenness Constraints. Combinatorics, Probability and Computing 19(5-6), 775–790 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goerdt, A., Falke, L.: Satisfiability thresholds beyond k−XORSAT. arXiv:cs/1112.2118Google Scholar
  17. 17.
    Hastad, J.: Some optimal inapproximability results. J. ACM 48, 798–859 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kolchin, V.F.: Random graphs and systems of linear equations in finite fields. Random Structures and Algorithms 5, 425–436 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Luby, M., Mitzenmacher, M., Shokrollahi, A., Spielman, D.A.: Efficient erasure coeds. IEEE Trans. Inform. Theory 47(2), 569–584 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Meisels, A., Shimony, S.E., Solotorevsky, G.: Bayes Networks for estimating the number of solutions to a CSP. In: Proceedings AAAI 1997, pp. 179–184 (1997)Google Scholar
  21. 21.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press (2005)Google Scholar
  22. 22.
    Molloy, M.: Cores in random hypergraphs and boolean formulas. Random Stuctures and Algorithms 27, 124–135 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Molloy, M.: Models for Random Constraint Satisfaction Problems. SIAM J. Comput. 32(4), 935–949 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Puyhaubert, V.: Generating functions and the satisfiability threshold. Discrete Mathematics and Theoretical Computer Science 6, 425–436 (2004)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Richardson, T.J., Urbanke, R.: Modern Coding Theory. Cambridge University Press (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Goerdt
    • 1
  • Lutz Falke
    • 1
  1. 1.Fakultät für InformatikTechnische Universität ChemnitzChemnitzGermany

Personalised recommendations