Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

Abstract

We study random constraint satisfaction problems (CSPs) in the unsatis able 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 satis ed 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 [Par80, Tal06, AC17], 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 [HS21] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms [CLSS22].