Sharp Thresholds for the Overlap Gap Property: Ising $p$-Spin Glass and Random $k$-SAT
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arXiv
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| Format: | Preprint |
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2023
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| author | Kızıldağ, Eren C. |
| author_facet | Kızıldağ, Eren C. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_09913 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp Thresholds for the Overlap Gap Property: Ising $p$-Spin Glass and Random $k$-SAT Kızıldağ, Eren C. Probability Computational Complexity Data Structures and Algorithms Mathematical Physics 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. |
| title | Sharp Thresholds for the Overlap Gap Property: Ising $p$-Spin Glass and Random $k$-SAT |
| topic | Probability Computational Complexity Data Structures and Algorithms Mathematical Physics |
| url | https://arxiv.org/abs/2309.09913 |