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Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3

Authors Evan Chang, Neel Kolhe, Youngtak Sohn

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LIPIcs.APPROX-RANDOM.2024.47.pdf
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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Random graphs
Keywords
  • Random constraint satisfaction problem
  • replica symmetry breaking
  • interpolation bound

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Abstract

For a large class of random constraint satisfaction problems (csp), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular k-nae-sat, random k-sat, and random regular k-sat for large enough k ≥ k₀ where k₀ is a large non-explicit constant. Establishing the same for small values of k ≥ 3 remains an important open problem in the study of random csps.
In this work, we study two closely related models of random csps, namely the 2-coloring on random d-regular k-uniform hypergraphs and the random d-regular k-nae-sat model. For every k ≥ 3, we prove that there is an explicit d_⋆(k) which gives a satisfiability upper bound for both of the models. Our upper bound d_⋆(k) for k ≥ 3 matches the prediction from statistical physics for the hypergraph 2-coloring by Dall’Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, d_⋆(k) coincides with the satisfiability threshold of random regular k-nae-sat for large enough k ≥ k₀ by Ding, Sly, Sun (2014).

Cite As Get BibTex

Evan Chang, Neel Kolhe, and Youngtak Sohn. Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.47

Author Details

Evan Chang
  • Massachusetts Institute of Technology, USA
Neel Kolhe
  • University of California, Berkeley, USA
Youngtak Sohn
  • Department of Mathematics, Massachusetts Institute of Technology, USA

Funding

  • Sohn, Youngtak: Supported by Simons-NSF Collaboration on Deep Learning NSF DMS-2031883 and Elchanan Mossel’s Vannevar Bush Faculty Fellowship award ONR-N00014-20-1-2826.

Acknowledgements

We thank the MIT PRIMES program and its organizers Pavel Etingof, Slava Gerovitch, and Tanya Khovanova for making this possible. Y.S. thanks Elchanan Mossel, Allan Sly, and Nike Sun for encouraging feedbacks.

References

  1. Dimitris Achlioptas, Arthur Chtcherba, Gabriel Istrate, and Cristopher Moore. The phase transition in 1-in-k SAT and NAE 3-sat. In Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '01, pages 721-722, Philadelphia, PA, USA, 2001. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=365411.365760.
  2. Dimitris Achlioptas and Cristopher Moore. On the 2-colorability of random hypergraphs. In José D. P. Rolim and Salil Vadhan, editors, Randomization and Approximation Techniques in Computer Science, pages 78-90, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. Google Scholar
  3. Dimitris Achlioptas and Cristopher Moore. Random k-SAT: two moments suffice to cross a sharp threshold. SIAM J. Comput., 36(3):740-762, 2006. URL: https://doi.org/10.1137/S0097539703434231.
  4. Dimitris Achlioptas and Assaf Naor. The two possible values of the chromatic number of a random graph. Ann. of Math. (2), 162(3):1335-1351, 2005. URL: https://doi.org/10.4007/annals.2005.162.1335.
  5. Dimitris Achlioptas, Assaf Naor, and Yuval Peres. Rigorous location of phase transitions in hard optimization problems. Nature, 435(7043):759-764, 2005. Google Scholar
  6. Dimitris Achlioptas and Yuval Peres. The threshold for random k-SAT is 2^klog 2-O(k). J. Amer. Math. Soc., 17(4):947-973, 2004. URL: https://doi.org/10.1090/S0894-0347-04-00464-3.
  7. N. Alon and Z. Bregman. Every 8-uniform 8-regular hypergraph is 2-colorable. Graphs and Combinatorics, 4(1):303-306, 1988. URL: https://doi.org/10.1007/BF01864169.
  8. Peter Ayre, Amin Coja-Oghlan, Pu Gao, and Noëla Müller. The satisfiability threshold for random linear equations. Combinatorica, 40(2):179-235, 2020. URL: https://doi.org/10.1007/s00493-019-3897-3.
  9. Peter Ayre, Amin Coja-Oghlan, and Catherine Greenhill. Lower bounds on the chromatic number of random graphs. Combinatorica, 42(5):617-658, 2022. URL: https://doi.org/10.1007/s00493-021-4236-z.
  10. Victor Bapst and Amin Coja-Oghlan. The condensation phase transition in the regular k-SAT model. In Approximation, randomization, and combinatorial optimization. Algorithms and techniques, volume 60 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 22, 18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  11. Victor Bapst, Amin Coja-Oghlan, Samuel Hetterich, Felicia Raßmann, and Dan Vilenchik. The condensation phase transition in random graph coloring. Comm. Math. Phys., 341(2):543-606, 2016. URL: https://doi.org/10.1007/s00220-015-2464-z.
  12. Béla Bollobás, Christian Borgs, Jennifer T. Chayes, Jeong Han Kim, and David B. Wilson. The scaling window of the 2-SAT transition. Random Structures Algorithms, 18(3):201-256, 2001. URL: https://doi.org/10.1002/rsa.1006.
  13. A. A. Borovkov. Generalization and refinement of the integro-local stone theorem for sums of random vectors. Theory of Probability & Its Applications, 61(4):590-612, 2017. URL: https://doi.org/10.1137/S0040585X97T988368.
  14. Evan Chang, Neel Kolhe, and Youngtak Sohn. Upper bounds on the 2-colorability threshold of random d-regular k-uniform hypergraphs for k ≥ 3. arXiv:2308.02075, 2023. Google Scholar
  15. V. Chvatal and B. Reed. Mick gets some (the odds are on his side) (satisfiability). In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, SFCS '92, pages 620-627, Washington, DC, USA, 1992. IEEE Computer Society. URL: https://doi.org/10.1109/SFCS.1992.267789.
  16. Amin Coja-Oghlan. Upper-bounding the k-colorability threshold by counting covers. Electron. J. Combin., 20(3):Paper 32, 28, 2013. Google Scholar
  17. Amin Coja-Oghlan, Charilaos Efthymiou, and Samuel Hetterich. On the chromatic number of random regular graphs. J. Combin. Theory Ser. B, 116:367-439, 2016. URL: https://doi.org/10.1016/j.jctb.2015.09.006.
  18. Amin Coja-Oghlan, Florent Krzakała, Will Perkins, and Lenka Zdeborová. Information-theoretic thresholds from the cavity method. Adv. Math., 333:694-795, 2018. URL: https://doi.org/10.1016/j.aim.2018.05.029.
  19. Amin Coja-Oghlan and Konstantinos Panagiotou. Catching the k-NAESAT threshold [extended abstract]. In STOC'12 - Proceedings of the 2012 ACM Symposium on Theory of Computing, pages 899-907. ACM, New York, 2012. URL: https://doi.org/10.1145/2213977.2214058.
  20. Amin Coja-Oghlan and Konstantinos Panagiotou. Going after the k-sat threshold. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC '13, pages 705-714, New York, NY, USA, 2013. Association for Computing Machinery. URL: https://doi.org/10.1145/2488608.2488698.
  21. Amin Coja-Oghlan and Konstantinos Panagiotou. The asymptotic k-SAT threshold. Adv. Math., 288:985-1068, 2016. URL: https://doi.org/10.1016/j.aim.2015.11.007.
  22. Amin Coja-Oghlan and Will Perkins. Spin systems on Bethe lattices. Communications in Mathematical Physics, 372(2):441-523, 2019. URL: https://doi.org/10.1007/s00220-019-03544-y.
  23. Amin Coja-Oghlan and Dan Vilenchik. Chasing the k-colorability threshold. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science - FOCS '13, pages 380-389. IEEE Computer Soc., Los Alamitos, CA, 2013. URL: https://doi.org/10.1109/FOCS.2013.48.
  24. Amin Coja-Oghlan and Lenka Zdeborová. The condensation transition in random hypergraph 2-coloring. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, pages 241-250. ACM, New York, 2012. Google Scholar
  25. L. Dall'Asta, A. Ramezanpour, and R. Zecchina. Entropy landscape and non-gibbs solutions in constraint satisfaction problems. Physical Review E, 77(3), March 2008. URL: https://doi.org/10.1103/physreve.77.031118.
  26. Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. URL: https://doi.org/10.1007/978-3-642-03311-7.
  27. Martin Dietzfelbinger, Andreas Goerdt, Michael Mitzenmacher, Andrea Montanari, Rasmus Pagh, and Michael Rink. Tight thresholds for cuckoo hashing via XORSAT. In Samson Abramsky, Cyril Gavoille, Claude Kirchner, Friedhelm Meyer auf der Heide, and Paul G. Spirakis, editors, Automata, Languages and Programming, pages 213-225, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. Google Scholar
  28. Jian Ding, Allan Sly, and Nike Sun. Satisfiability threshold for random regular nae-sat. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 814-822, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2591796.2591862.
  29. Jian Ding, Allan Sly, and Nike Sun. Proof of the satisfiability conjecture for large k. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 59-68, New York, NY, USA, 2015. ACM. URL: https://doi.org/10.1145/2746539.2746619.
  30. Jian Ding, Allan Sly, and Nike Sun. Maximum independent sets on random regular graphs. Acta Math., 217(2):263-340, 2016. URL: https://doi.org/10.1007/s11511-017-0145-9.
  31. Jian Ding, Allan Sly, and Nike Sun. Satisfiability threshold for random regular NAE-SAT. Commun. Math. Phys., 341(2):435-489, 2016. Google Scholar
  32. Jian Ding, Allan Sly, and Nike Sun. Proof of the satisfiability conjecture for large k. Annals of Mathematics, 196(1):1-388, 2022. URL: https://doi.org/10.4007/annals.2022.196.1.1.
  33. Olivier Dubois and Jacques Mandler. The 3-XORSAT threshold. In Proceedings of the 43rd Symposium on Foundations of Computer Science, FOCS '02, pages 769-778, Washington, DC, USA, 2002. IEEE Computer Society. URL: http://dl.acm.org/citation.cfm?id=645413.652160.
  34. Martin Dyer, Alan Frieze, and Catherine Greenhill. On the chromatic number of a random hypergraph. Journal of Combinatorial Theory, Series B, 113:68-122, 2015. URL: https://doi.org/10.1016/j.jctb.2015.01.002.
  35. Silvio Franz and Michele Leone. Replica bounds for optimization problems and diluted spin systems. Journal of Statistical Physics, 111(3):535-564, 2003. URL: https://doi.org/10.1023/A:1022885828956.
  36. Yuzhou Gu and Yury Polyanskiy. Uniqueness of bp fixed point for the potts model and applications to community detection. arXiv preprint, arXiv:2303.14688, 2023. Google Scholar
  37. Francesco Guerra. Broken replica symmetry bounds in the mean field spin glass model. Communications in Mathematical Physics, 233(1):1-12, 2003. URL: https://doi.org/10.1007/s00220-002-0773-5.
  38. Michael A. Henning and Anders Yeo. 2-colorings in k-regular k-uniform hypergraphs. European Journal of Combinatorics, 34(7):1192-1202, 2013. URL: https://doi.org/10.1016/j.ejc.2013.04.005.
  39. Michael A. Henning and Anders Yeo. Not-all-equal 3-sat and 2-colorings of 4-regular 4-uniform hypergraphs. Discrete Mathematics, 341(8):2285-2292, 2018. URL: https://doi.org/10.1016/j.disc.2018.05.002.
  40. Svante Janson, Tomasz Luczak, and Andrzej Rucinski. Random graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000. URL: https://doi.org/10.1002/9781118032718.
  41. Richard M. Karp. Reducibility among Combinatorial Problems, pages 85-103. Springer US, Boston, MA, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  42. Lefteris M. Kirousis, Evangelos Kranakis, Danny Krizanc, and Yannis C. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures Algorithms, 12(3):253-269, 1998. URL: https://doi.org/10.1002/(SICI)1098-2418(199805)12:3<253::AID-RSA3>3.3.CO;2-H.
  43. Florent Krzakała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25):10318-10323, 2007. URL: https://doi.org/10.1073/pnas.0703685104.
  44. Marc Lelarge and Mendes Oulamara. Replica bounds by combinatorial interpolation for diluted spin systems. Journal of Statistical Physics, 173(3–4):917-940, February 2018. URL: https://doi.org/10.1007/s10955-018-1964-6.
  45. Stephan Mertens, Marc Mézard, and Riccardo Zecchina. Threshold values of random k-sat from the cavity method. Random Structures & Algorithms, 28(3):340-373, 2006. URL: https://doi.org/10.1002/rsa.20090.
  46. M. Mézard, G. Parisi, and R. Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, 297(5582):812-815, 2002. URL: https://doi.org/10.1126/science.1073287.
  47. Marc Mézard and Andrea Montanari. Information, physics, and computation. Oxford Graduate Texts. Oxford University Press, Oxford, 2009. URL: https://doi.org/10.1093/acprof:oso/9780198570837.001.0001.
  48. Andrea Montanari, Feng Ruan, Youngtak Sohn, and Jun Yan. The generalization error of max-margin linear classifiers: Benign overfitting and high-dimensional asymptotics in the overparametrized regime. arXiv, 2019. URL: https://arxiv.org/abs/1911.01544.
  49. Danny Nam, Allan Sly, and Youngtak Sohn. One-step replica symmetry breaking of random regular NAE-SAT I. arXiv preprint, 2020. URL: https://arxiv.org/abs/2011.14270.
  50. Danny Nam, Allan Sly, and Youngtak Sohn. One-step replica symmetry breaking of random regular NAE-SAT. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 310-318, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00039.
  51. Danny Nam, Allan Sly, and Youngtak Sohn. One-step replica symmetry breaking of random regular NAE-SAT II. Communications in Mathematical Physics, 405(3):61, 2024. URL: https://doi.org/10.1007/s00220-023-04868-6.
  52. Dmitry Panchenko and Michel Talagrand. Bounds for diluted mean-fields spin glass models. Probability Theory and Related Fields, 130(3):319-336, 2004. URL: https://doi.org/10.1007/s00440-004-0342-2.
  53. Boris Pittel and Gregory B. Sorkin. The satisfiability threshold for k-XORSAT. Combin. Probab. Comput., 25(2):236-268, 2016. URL: https://doi.org/10.1017/S0963548315000097.
  54. P. D. Seymour. On the two-coloring of hypergraphs. The Quarterly Journal of Mathematics, 25(1):303-311, January 1974. URL: https://doi.org/10.1093/qmath/25.1.303.
  55. Mariya Shcherbina and Brunello Tirozzi. Rigorous solution of the Gardner problem. Communications in Mathematical Physics, 234(3):383-422, 2003. Google Scholar
  56. Allan Sly and Youngtak Sohn. Local geometry of NAE-SAT solutions in the condensation regime. arXiv preprint, 2023. URL: https://arxiv.org/abs/2305.17334.
  57. Allan Sly, Nike Sun, and Yumeng Zhang. The number of solutions for random regular NAE-SAT. In Proceedings of the 57th Symposium on Foundations of Computer Science, FOCS '16, pages 724-731, 2016. Google Scholar
  58. Allan Sly, Nike Sun, and Yumeng Zhang. The number of solutions for random regular NAE-SAT. Probability Theory and Related Fields, 182(1-2):1-109, 2022. URL: https://doi.org/10.1007/s00440-021-01029-5.
  59. Michel Talagrand. Mean Field Models for Spin Glasses: Volume I. Springer-Verlag, Berlin, 2010. Google Scholar
  60. Qian Yu and Yury Polyanskiy. Ising model on locally tree-like graphs: Uniqueness of solutions to cavity equations. arXiv preprint, 2022. URL: https://arxiv.org/abs/2211.15242.
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