Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

TL;DR

This paper connects the structure of near-optimal solutions in random Max-CSPs to spin glass models, revealing that algorithmic hardness in Max-CSPs mirrors that of spin glasses through the overlap gap property.

Contribution

It establishes a theoretical link between Max-CSPs and spin glasses, enabling the transfer of hardness results and providing numerical estimates for Max-CSP satisfiability thresholds.

Findings

Max-CSP satisfiability relates to spin glass ground state energy via the Parisi formula.

Max-CSPs exhibit the overlap gap property if and only if their spin glass counterparts do.

Results obstruct classical and quantum algorithms based on overlap concentration.

Abstract

We study random constraint satisfaction problems (CSPs) in the unsatisfiable regime. We relate the structure of near-optimal solutions for any 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 prove two main consequences: 1) We relate the maximum fraction of constraints that can be satisfied in a random Max-CSP to the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula, we provide numerical values for some popular CSPs. 2) We prove that a Max-CSP 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 et al. [arXiv:2110.07847, 2021] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms.