Advertisement

Unsatisfiability Bounds for Random CSPs from an Energetic Interpolation Method

  • Dimitris Achlioptas
  • Ricardo Menchaca-Mendez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

The interpolation method, originally developed in statistical physics, transforms distributions of random CSPs to distributions of much simpler problems while bounding the change in a number of associated statistical quantities along the transformation path. After a number of further mathematical developments, it is now known that, in principle, the method can yield rigorous unsatisfiability bounds if one “plugs in an appropriate functional distribution”. A drawback of the method is that identifying appropriate distributions and plugging them in leads to major analytical challenges as the distributions required are, in fact, infinite dimensional objects. We develop a variant of the interpolation method for random CSPs on arbitrary sparse degree distributions which trades accuracy for tractability. In particular, our bounds only require the solution of a 1-dimensional optimization problem (which typically turns out to be very easy) and as such can be used to compute explicit rigorous unsatisfiability bounds.

Keywords

Energy Function Interpolation Method Univariate Factor Degree Sequence Optimal Assignment 
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.

References

  1. 1.
    Achioptas, D., Sorkin, G.: Optimal myopic algorithms for random 3-sat. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science 2000, pp. 590–600. IEEE (2000)Google Scholar
  2. 2.
    Achlioptas, D.: Lower bounds for random 3-sat via differential equations. Theoretical Computer Science 265(1-2), 159–185 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Achlioptas, D., Beame, P., Molloy, M.: A sharp threshold in proof complexity yields lower bounds for satisfiability search. Journal of Computer and System Sciences 68(2), 238–268 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Achlioptas, D., Coja-Oghlan, A.: Algorithmic barriers from phase transitions. In: 49th Annual IEEE Symp. on Foundations of Computer Science 2008, pp. 793–802 (2008)Google Scholar
  5. 5.
    Achlioptas, D., Menchaca-Mendez, R.: Exponential lower bounds for dpll algorithms on satisfiable random 3-cnf formulas (2012) (to appear in SAT 2012)Google Scholar
  6. 6.
    Bayati, M., Gamarnik, D., Tetali, P.: Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. In: STOC 2010, pp. 105–114 (2010)Google Scholar
  7. 7.
    Chvatal, V., Szemeredi, E.: Many hard examples for resolution. Journal of the Association for Computing Machinery 35(4), 759–768 (1988)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Coja-Oghlan, A.: On belief propagation guided decimation for random k-sat. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 957–966. SIAM (2011)Google Scholar
  9. 9.
    Díaz, J., Kirousis, L., Mitsche, D., Pérez-Giménez, X.: On the satisfiability threshold of formulas with three literals per clause. Theoretical Computer Science 410(30-32), 2920–2934 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. Journal of Statistical Physics 111(3), 535–564 (2003)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Franz, S., Leone, M., Toninelli, F.: Replica bounds for diluted non-poissonian spin systems. Journal of Physics A: Mathematical and General 36, 10967 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Guerra, F., Toninelli, F.: The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics 230(1), 71–79 (2002)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kaporis, A., Kirousis, L., Lalas, E.: The probabilistic analysis of a greedy satisfiability algorithm. Random Structures & Algorithms 28(4), 444–480 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mertens, S., Mézard, M., Zecchina, R.: Threshold values of random k-sat from the cavity method. Random Structures & Algorithms 28(3), 340–373 (2006)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Mezard, M., Montanari, A.: Information, physics, and computation. Oxford University Press, USA (2009)MATHCrossRefGoogle Scholar
  16. 16.
    Monasson, R., Zecchina, R.: Tricritical points in random combinatorics: the-sat case. Journal of Physics A: Mathematical and General 31, 9209 (1998)MATHCrossRefGoogle Scholar
  17. 17.
    Monasson, R., Zecchina, R.: Entropy of the K -satisfiability problem. Phys. Rev. Lett. 76, 3881–3885 (1996), http://link.aps.org/doi/10.1103/PhysRevLett.76.3881 MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Monasson, R., Zecchina, R.: Statistical mechanics of the random k-satisfiability model. Phys. Rev. E 56, 1357–1370 (1997), http://link.aps.org/doi/10.1103/PhysRevE.56.1357 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Montanari, A.: Tight bounds for ldpc and ldgm codes under map decoding. IEEE Transactions on Information Theory 51(9), 3221–3246 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Panchenko, D., Talagrand, M.: Bounds for diluted mean-fields spin glass models. Probability Theory and Related Fields 130(3), 319–336 (2004)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dimitris Achlioptas
    • 1
    • 2
    • 3
  • Ricardo Menchaca-Mendez
    • 3
  1. 1.University of AthensGreece
  2. 2.CTIGreece
  3. 3.University of CaliforniaSanta CruzUSA

Personalised recommendations