Beyond-mean-field fluctuations for the solution of constraint satisfaction problems

Abstract

Constraint Satisfaction Problems (CSPs) lie at the heart of complexity theory and find application in a plethora of prominent tasks ranging from cryptography to genetics. Classical approaches use Hopfield networks to find approximate solutions while recently, modern machine-learning techniques like graph neural networks have become popular for this task. In this study, we employ the known mapping of MAX-2-SAT, a class of CSPs, to a spin-glass system from statistical physics, and use Glauber dynamics to approximately find its ground state, which corresponds to the optimal solution of the underlying problem. We show that Glauber dynamics outperforms the traditional Hopfield-network approach and can compete with state-of-the-art solvers. A systematic theoretical analysis uncovers the role of stochastic fluctuations in finding CSP solutions: even in the absence of thermal fluctuations at $T=0$ a significant portion of spins, which correspond to the CSP variables, attains an effective spin-dependent non-zero temperature. These spins form a subspace in which the stochastic Glauber dynamics continuously performs flips to eventually find better solutions. This is possible since the energy is degenerate, such that spin flips in this free-spin space do not require energy. Our theoretical analysis leads to deterministic solvers that effectively account for such fluctuations, thereby reaching state-of-the-art performance.