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Sharp Thresholds for the Overlap Gap Property: Ising p-Spin Glass and Random k-SAT

Author Eren C. Kızıldağ

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ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Combinatorial optimization
Keywords
  • spin glasses
  • p-spin model
  • random constraint satisfaction problems
  • overlap gap property
  • phase transitions
  • computational complexity

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Abstract

The Ising p-spin glass and random k-SAT are two canonical examples of disordered systems that play a central role in understanding the link between geometric features of optimization landscapes and computational tractability. Both models exhibit hard regimes where all known polynomial-time algorithms fail and possess the multi Overlap Gap Property (m-OGP), an intricate geometrical property that rigorously rules out a broad class of algorithms exhibiting input stability. 
We establish that, in both models, the symmetric m-OGP undergoes a sharp phase transition, and we pinpoint its exact threshold. For the Ising p-spin glass, our results hold for all sufficiently large p; for the random k-SAT, they apply to all k growing mildly with the number of Boolean variables. Notably, our findings yield qualitative insights into the power of OGP-based arguments. A particular consequence for the Ising p-spin glass is that the strength of the m-OGP in establishing algorithmic hardness grows without bound as m increases. 
These are the first sharp threshold results for the m-OGP. Our analysis hinges on a judicious application of the second moment method, enhanced by concentration. While a direct second moment calculation fails, we overcome this via a refined approach that leverages an argument of Frieze [Frieze, 1990] and exploiting concentration properties of carefully constructed random variables.

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Eren C. Kızıldağ. Sharp Thresholds for the Overlap Gap Property: Ising p-Spin Glass and Random k-SAT. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.48

Author Details

Eren C. Kızıldağ
  • Department of Statistics, University of Illinois Urbana-Champaign, IL, USA

Acknowledgements

I am grateful to David Gamarnik for his guidance and many helpful discussions during the early stages of this work. I would like to thank Brice Huang for carefully proofreading the draft and offering insightful comments, Aukosh Jagannath for numerous valuable discussions regarding spin glasses, Lutz Warnke for a helpful discussion on the bounded differences inequality, and anonymous referees for their valuable feedback.

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