Local Geometry of NAE-SAT Solutions in the Condensation Regime
Authors: 
Allan Sly, Youngtak SohnAuthors Info & Claims
Allan Sly, Youngtak SohnAuthors Info & ClaimsPages 1083 - 1093
Abstract
The local behavior of typical solutions of random constraint satisfaction problems (csp) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos, M'uller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular nae-sat model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters. This is the first description of the local weak limit in the condensation regime for any sparse random csps in the one-step replica symmetry breaking (1rsb) class. Our result is non-asymptotic, and characterizes the tight fluctuation O(n−1/2) around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1rsb belief-propagation fixed point. We believe that our proof technique has broad applicability to random csps in the 1rsb class.
References
[1]
Dimitris Achlioptas and Amin Coja-Oghlan. 2008. Algorithmic barriers from phase transitions. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science. 793–802.
[2]
Dimitris Achlioptas, Assaf Naor, and Yuval Peres. 2005. Rigorous location of phase transitions in hard optimization problems. Nature, 435, 7043 (2005), 759–764.
[3]
David J Aldous. 2001. The ζ (2) limit in the random assignment problem. Random Structures & Algorithms, 18, 4 (2001), 381–418.
[4]
Itai Benjamini and Oded Schramm. 2001. Recurrence of distributional limits of finite planar graphs. Electronic Journal of Probability [electronic only], 6 (2001).
[5]
Charles Bordenave and Pietro Caputo. 2015. Large deviations of empirical neighborhood distribution in sparse random graphs. Probability Theory and Related Fields, 163, 1 (2015), 149–222. https://doi.org/10.1007/s00440-014-0590-8
[6]
A. A. Borovkov. 2017. Generalization and Refinement of the Integro-Local Stone Theorem for Sums of Random Vectors. Theory of Probability & Its Applications, 61, 4 (2017), 590–612. https://doi.org/10.1137/S0040585X97T988368 arxiv:https://doi.org/10.1137/S0040585X97T988368.
[7]
Amin Coja-Oghlan. 2014. The Asymptotic K-SAT Threshold. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing (STOC ’14). Association for Computing Machinery, New York, NY, USA. 804–813. isbn:9781450327107 https://doi.org/10.1145/2591796.2591822
[8]
Amin Coja-Oghlan, Charilaos Efthymiou, and Nor Jaafari. 2018. Local convergence of random graph colorings. Combinatorica, 38, 2 (2018), 341–380.
[9]
Amin Coja-Oghlan, Tobias Kapetanopoulos, and Noela Müller. 2020. The replica symmetric phase of random constraint satisfaction problems. Combinatorics, Probability and Computing, 29, 3 (2020), 346–422.
[10]
Amin Coja-Oghlan and Konstantinos Panagiotou. 2013. Going after the K-SAT Threshold. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing (STOC ’13). Association for Computing Machinery, New York, NY, USA. 705–714. isbn:9781450320290 https://doi.org/10.1145/2488608.2488698
[11]
Jian Ding, Allan Sly, and Nike Sun. 2016. Satisfiability threshold for random regular NAE-SAT. Commun. Math. Phys., 341, 2 (2016), 435–489.
[12]
Jian Ding, Allan Sly, and Nike Sun. 2022. Proof of the satisfiability conjecture for large k. Annals of Mathematics, 196, 1 (2022), 1 – 388. https://doi.org/10.4007/annals.2022.196.1.1
[13]
Antoine Gerschenfeld and Andrea Montanari. 2007. Reconstruction for models on random graphs. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS’07). 194–204.
[14]
Svante Janson. 1995. Random regular graphs: asymptotic distributions and contiguity. Combinatorics, Probability and Computing, 4, 4 (1995), 369–405.
[15]
Florent Krz̧akała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. 2007. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104, 25 (2007), 10318–10323. issn:0027-8424 https://doi.org/10.1073/pnas.0703685104 arxiv:https://www.pnas.org/content/104/25/10318.full.pdf.
[16]
Marc Mézard and Andrea Montanari. 2009. Information, physics, and computation. Oxford University Press. isbn:9780198570837 https://doi.org/10.1093/acprof:oso/9780198570837.001.0001
[17]
M. Mézard, G. Parisi, and R. Zecchina. 2002. Analytic and Algorithmic Solution of Random Satisfiability Problems. Science, 297, 5582 (2002), 812–815. issn:0036-8075 https://doi.org/10.1126/science.1073287 arxiv:https://science.sciencemag.org/content/297/5582/812.full.pdf.
[18]
Michael Molloy. 2018. The freezing threshold for k-colourings of a random graph. Journal of the ACM (JACM), 65, 2 (2018), 1–62.
[19]
Michael Molloy and Ricardo Restrepo. 2013. Frozen Variables in Random Boolean Constraint Satisfaction Problems. In Proceedings of the Twenty-fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’13). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA. 1306–1318. isbn:978-1-611972-51-1 http://dl.acm.org/citation.cfm?id=2627817.2627912
[20]
Andrea Montanari, Elchanan Mossel, and Allan Sly. 2012. The weak limit of Ising models on locally tree-like graphs. Probability Theory and Related Fields, 152 (2012), 31–51.
[21]
Andrea Montanari, Ricardo Restrepo, and Prasad Tetali. 2011. Reconstruction and clustering in random constraint satisfaction problems. SIAM Journal on Discrete Mathematics, 25, 2 (2011), 771–808.
[22]
Andrea Montanari, Federico Ricci-Tersenghi, and Guilhem Semerjian. 2008. Clusters of solutions and replica symmetry breaking in random k-satisfiability. Journal of Statistical Mechanics: Theory and Experiment, 2008, 04 (2008), P04004.
[23]
Elchanan Mossel, Joe Neeman, and Allan Sly. 2015. Reconstruction and estimation in the planted partition model. Probability Theory and Related Fields, 162 (2015), 431–461.
[24]
Elchanan Mossel, Allan Sly, and Youngtak Sohn. 2023. Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing (STOC 2023). Association for Computing Machinery, New York, NY, USA. 96–102. isbn:9781450399135 https://doi.org/10.1145/3564246.3585155
[25]
Danny Nam, Allan Sly, and Youngtak Sohn. 2020. One-step replica symmetry breaking of random regular NAE-SAT I. arXiv preprint, arXiv:2011.14270.
[26]
Danny Nam, Allan Sly, and Youngtak Sohn. 2022. One-step replica symmetry breaking of random regular NAE-SAT. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). 310–318. https://doi.org/10.1109/FOCS52979.2021.00039
[27]
Danny Nam, Allan Sly, and Youngtak Sohn. 2024. One-Step Replica Symmetry Breaking of Random Regular NAE-SAT II. Communications in Mathematical Physics, 405, 3 (2024), 61. isbn:1432-0916 https://doi.org/10.1007/s00220-023-04868-6
[28]
Dmitry Panchenko. 2013. The Sherrington-Kirkpatrick model. Springer, New York. isbn:978-1-4614-6288-0; 978-1-4614-6289-7 https://doi.org/10.1007/978-1-4614-6289-7
[29]
Giorgio Parisi. 2002. On local equilibrium equations for clustering states. arXiv:cs/0212047.
[30]
Robert W. Robinson and Nicholas C. Wormald. 1994. Almost all regular graphs are Hamiltonian. Random Structures & Algorithms, 5, 2 (1994), 363–374.
[31]
Allan Sly and Youngtak Sohn. 2023. Local geometry of NAE-SAT solutions in the condensation regime. arXiv preprint, arXiv:2305.17334.
[32]
Allan Sly, Nike Sun, and Yumeng Zhang. 2022. The number of solutions for random regular NAE-SAT. Probability Theory and Related Fields, 182, 1 (2022), 1–109. https://doi.org/10.1007/s00440-021-01029-5
[33]
Lenka Zdeborová and Florent Krzakał a. 2007. Phase transitions in the coloring of random graphs. Physical Review E, 76, 3 (2007), 031131.
Index Terms
- Local Geometry of NAE-SAT Solutions in the Condensation Regime
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June 2024
2049 pages
ISBN:9798400703836
DOI:10.1145/3618260
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Published: 11 June 2024
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