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

TL;DR

This paper establishes the precise phase transition thresholds for the multi Overlap Gap Property in Ising p-spin glasses and random k-SAT models, revealing their implications for computational hardness and algorithmic limitations.

Contribution

It provides the first sharp threshold results for the m-OGP in these models, demonstrating how the property correlates with algorithmic difficulty and grows stronger with m.

Findings

Sharp phase transition thresholds for m-OGP in both models

The strength of m-OGP increases with m, indicating greater algorithmic hardness

Application of refined second moment method with concentration techniques

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~\cite{frieze1990independence} and exploiting concentration properties of carefully constructed random variables.