Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems

TL;DR

This paper studies spatially coupled constraint satisfaction problems, showing that coupling affects certain thresholds but not the fundamental SAT-UNSAT transition, and explores implications for algorithmic bounds.

Contribution

It proves that the SAT-UNSAT transition remains unchanged by spatial coupling and analyzes how other thresholds behave, providing insights into algorithmic improvements.

Findings

SAT-UNSAT threshold unaffected by coupling

Survey propagation threshold increases and saturates with coupling range

Dynamic threshold exhibits saturation towards condensation threshold

Abstract

We consider chains of random constraint satisfaction models that are spatially coupled across a finite window along the chain direction. We investigate their phase diagram at zero temperature using the survey propagation formalism and the interpolation method. We prove that the SAT-UNSAT phase transition threshold of an infinite chain is identical to the one of the individual standard model, and is therefore not affected by spatial coupling. We compute the survey propagation complexity using population dynamics as well as large degree approximations, and determine the survey propagation threshold. We find that a clustering phase survives coupling. However, as one increases the range of the coupling window, the survey propagation threshold increases and saturates towards the phase transition threshold. We also briefly discuss other aspects of the problem. Namely, the condensation threshold is not affected by coupling, but the dynamic threshold displays saturation towards the condensation one. All these features may provide a new avenue for obtaining better provable algorithmic lower bounds on phase transition thresholds of the individual standard model.