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Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

Authors Chris Jones , Kunal Marwaha , Juspreet Singh Sandhu , Jonathan Shi

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Author Details

Chris Jones
  • University of Chicago, Il, USA
Kunal Marwaha
  • University of Chicago, USA
Juspreet Singh Sandhu
  • Harvard University, Cambridge, MA, USA
Jonathan Shi
  • Bocconi University, Milano, Italy


We thank Antares Chen for collaborating during the early stages of this project. We also thank Peter J. Love for helpful feedback on a previous version of this manuscript. KM thanks Ryan Robinett for a tip on improving the numerical integration scheme. JSS did some of this work as a visiting student at Bocconi University. We thank the anonymous reviewers for many suggestions to improve the text, for pointing out an error in a proof, and for the reference [Barbier and Panchenko, 2022].

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Chris Jones, Kunal Marwaha, Juspreet Singh Sandhu, and Jonathan Shi. Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 77:1-77:26, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We study random constraint satisfaction problems (CSPs) at large clause density. We relate the structure of near-optimal solutions for any Boolean Max-CSP to that for an associated spin glass on the hypercube, using the Guerra-Toninelli interpolation from statistical physics. The noise stability polynomial of the CSP’s predicate is, up to a constant, the mixture polynomial of the associated spin glass. We show two main consequences: 1) We prove that the maximum fraction of constraints that can be satisfied in a random Max-CSP at large clause density is determined by the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula [Parisi, 1980; Talagrand, 2006; Auffinger and Chen, 2017], we provide numerical values for some popular CSPs. 2) We prove that a Max-CSP at large clause density possesses generalized versions of the overlap gap property if and only if the same holds for the corresponding spin glass. We transfer results from [Huang and Sellke, 2021] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms [Chou et al., 2022].

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Probability and statistics
  • spin glass
  • overlap gap property
  • constraint satisfaction problem
  • Guerra-Toninelli interpolation


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