Document Open Access Logo

The Number of Random 2-SAT Solutions Is Asymptotically Log-Normal

Authors Arnab Chatterjee, Amin Coja-Oghlan, Noela Müller, Connor Riddlesden, Maurice Rolvien, Pavel Zakharov, Haodong Zhu

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2024.39.pdf
  • Filesize: 2.61 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • satisfiability problem
  • 2-SAT
  • random satisfiability
  • central limit theorem

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order √n, with n the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are bounded throughout all or most of the satisfiable regime.

Cite As Get BibTex

Arnab Chatterjee, Amin Coja-Oghlan, Noela Müller, Connor Riddlesden, Maurice Rolvien, Pavel Zakharov, and Haodong Zhu. The Number of Random 2-SAT Solutions Is Asymptotically Log-Normal. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.39

Author Details

Arnab Chatterjee
  • Department of Computer Science, TU Dortmund, Germany
Amin Coja-Oghlan
  • Department of Computer Science, TU Dortmund, Germany
Noela Müller
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, Netherlands
Connor Riddlesden
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, Netherlands
Maurice Rolvien
  • Department of Computer Science, TU Dortmund, Germany
Pavel Zakharov
  • Department of Computer Science, TU Dortmund, Germany
Haodong Zhu
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, Netherlands

Funding

  • Coja-Oghlan, Amin: DFG CO 646/3, DFG CO 646/5 and DFG CO 646/6.
  • Müller, Noela: NWO Gravitation grant NETWORKS-024.002.003.
  • Zakharov, Pavel: DFG CO 646/6
  • Zhu, Haodong: European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 945045 and NWO Gravitation project NETWORKS-024.002.003

References

  1. E. Abbe, S. Li, and A. Sly. Proof of the contiguity conjecture and lognormal limit for the symmetric perceptron. In Proc. 62nd FOCS, pages 327-338, 2022. Google Scholar
  2. E. Abbe and A. Montanari. On the concentration of the number of solutions of random satisfiability formulas. RSA, 45:362-382, 2014. Google Scholar
  3. D. Achlioptas, A. Chtcherba, G. Istrate, and C. Moore. The phase transition in 1-in-k sat and nae 3-sat. In Proc. 12th SODA, pages 721-722, 2001. Google Scholar
  4. D. Achlioptas and A. Coja-Oghlan. Algorithmic barriers from phase transitions. In Proc. 49th FOCS, pages 793-802, 2008. Google Scholar
  5. D. Achlioptas, A. Coja-Oghlan, M. Hahn-Klimroth, J. Lee, N. Müller, M. Penschuck, and G. Zhou. The number of satisfying assignments of random 2-sat formulas. Random Structures and Algorithms, 58:609-647, 2021. Google Scholar
  6. D. Achlioptas and C. Moore. Random k-sat: two moments suffice to cross a sharp threshold. SIAM Journal on Computing, 36:740-762, 2006. Google Scholar
  7. D. Achlioptas, A. Naor, and Y. Peres. Rigorous location of phase transitions in hard optimization problems. Nature, 435:759-764, 2005. Google Scholar
  8. D. Achlioptas and Y. Peres. The threshold for random k-sat is 2^k ln 2 - O(k). Journal of the AMS, 17:947-973, 2004. Google Scholar
  9. D. Aldous and J. Steele. The objective method: probabilistic combinatorial optimization and local weak convergence. In H. Kesten, editor, Probability on Discrete Structures. Springer, 2004. Google Scholar
  10. B. Aspvall, M. F. Plass, and R. E. Tarjan. A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters, 8:121-123, 1979. Google Scholar
  11. P. Ayre, A. Coja-Oghlan, P. Gao, and N. Müller. The satisfiability threshold for random linear equations. Combinatorica, 40:179-235, 2020. Google Scholar
  12. V. Bapst, A. Coja-Oghlan, and C. Efthymiou. Planting colourings silently. Combinatorics, Probability and Computing, 26:338-366, 2017. Google Scholar
  13. V. Bapst, A. Coja-Oghlan, S. Hetterich, F. Rassmann, and D. Vilenchik. The condensation phase transition in random graph coloring. Communications in Mathematical Physics, 341:543-606, 2016. Google Scholar
  14. B. Bollobás, C. Borgs, J. Chayes, J. Kim, and D. Wilson. The scaling window of the 2-sat transition. RSA, 18:201-256, 2001. Google Scholar
  15. G. Bresler and B. Huang. The algorithmic phase transition of random k-sat for low degree polynomials. In Proc. 62nd FOCS, pages 298-309, 2021. Google Scholar
  16. S. Cao. Central limit theorems for combinatorial optimization problems on sparse erdős-rényi graphs. Annals of Applied Probability, 31:1687-1723, 2021. Google Scholar
  17. P. Cheeseman, B. Kanefsky, and W. Taylor. Where the really hard problems are. In Proc. IJCAI, pages 331-337, 1991. Google Scholar
  18. W.-K. Chen, P. Dey, and D. Panchenko. Fluctuations of the free energy in the mixed p-spin models with external field. Probability Theory and Related Fields, 168:41-53, 2017. Google Scholar
  19. V. Chvátal and B. Reed. Mick gets some (the odds are on his side). In Proc. 33th FOCS, pages 620-627, 1992. Google Scholar
  20. A. Coja-Oghlan, T. Kapetanopoulos, and N. Müller. The replica symmetric phase of random constraint satisfaction problems. Combinatorics, Probability and Computing, 29:346-422, 2020. Google Scholar
  21. A. Coja-Oghlan, F. Krzakala, W. Perkins, and L. Zdeborová. Information-theoretic thresholds from the cavity method. Advances in Mathematics, 333:694-795, 2018. Google Scholar
  22. A. Coja-Oghlan and K. Panagiotou. The asymptotic k-sat threshold. Advances in Mathematics, 288:985-1068, 2016. Google Scholar
  23. A. Coja-Oghlan and N. Wormald. The number of satisfying assignments of random regular k-sat formulas. Combinatorics, Probability and Computing, 27:496-530, 2018. Google Scholar
  24. J. Ding, A. Sly, and N. Sun. Proof of the satisfiability conjecture for large k. Annals of Mathematics, 196:1-388, 2022. Google Scholar
  25. S. Dovgal, É. de Panafieu, and V. Ravelomanana. Exact enumeration of satisfiable 2-sat formulae. arXiv:2108.08067, 2021. URL: https://arxiv.org/abs/2108.08067.
  26. O. Dubois and J. Mandler. The 3-xorsat threshold. In Proc. 43rd FOCS, pages 769-778, 2002. Google Scholar
  27. G. Eagleson. Martingale convergence to mixtures of infinitely divisible laws. Annals of Probability, 3:557-562, 1975. Google Scholar
  28. C. Efthymiou. On sampling symmetric gibbs distributions on sparse random graphs and hypergraphs. In Proc. 49th ICALP, page 57, 2022. Google Scholar
  29. E. Friedgut. Sharp thresholds of graph properties, and the k-sat problem. Journal of the AMS, 12:1017-1054, 1999. Google Scholar
  30. M. Glasgow, M. Kwan, A. Sah, and M. Sawhney. A central limit theorem for the matching number of a sparse random graph. arXiv:2402.05851, 2024. Google Scholar
  31. A. Goerdt. A threshold for unsatisfiability. Journal of Computer and System Sciences, 53:469-486, 1996. Google Scholar
  32. E. Kreačič. Some problems related to the Karp-Sipser algorithm on random graphs. PhD thesis, University of Oxford, 2017. Google Scholar
  33. F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104:10318-10323, 2007. Google Scholar
  34. M. Mézard and A. Montanari. Information, physics and computation. Oxford University Press, 2009. Google Scholar
  35. M. Mézard, G. Parisi, and R. Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, 297:812-815, 2002. Google Scholar
  36. R. Monasson and R. Zecchina. The entropy of the k-satisfiability problem. Physical Review Letters, 76:3881, 1996. Google Scholar
  37. A. Montanari and D. Shah. Counting good truth assignments of random k-sat formulae. In Proc. 18th SODA, pages 1255-1264, 2007. Google Scholar
  38. D. Panchenko. On the replica symmetric solution of the K-sat model. Electronic Journal of Probability, 19:67, 2014. Google Scholar
  39. D. Panchenko and M. Talagrand. Bounds for diluted mean-fields spin glass models. Probability Theory and Related Fields, 130:319-336, 2004. Google Scholar
  40. B. Pittel and G. Sorkin. The satisfiability threshold for k-xorsat. Combinatorics, Probability and Computing, 25:236-268, 2016. Google Scholar
  41. F. Rassmann. On the number of solutions in random graph k-colouring. Combinatorics, Probability and Computing, 28:130-158, 2019. Google Scholar
  42. R. Robinson and N. Wormald. Almost all regular graphs are hamiltonian. Random Structures and Algorithms, 5:363-374, 1994. Google Scholar
  43. A. Sly, N. Sun, and Y. Zhang. The number of solutions for random regular nae-sat. Probability Theory and Related Fields, 182:1-109, 2022. Google Scholar
  44. M. Talagrand. The high temperature case for the random K-sat problem. Probability Theory and Related Fields, 119:187-212, 2001. Google Scholar
  45. L. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8:410-421, 1979. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact