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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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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.

Cite AsGet 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

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).

Subject Classification

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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