Quiet Planting in the Locked Constraint Satisfaction Problems

We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles. The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to be crucial. Our main result is the location of the hard region in the planted ensemble. In a part of that hard region instances have with high probability a single satisfying assignment.

💡 Research Summary

The paper investigates the planted ensemble of locked constraint satisfaction problems (CSPs), a class of CSPs in which each constraint “locks’’ the variables so that any satisfying assignment is highly constrained and typically isolated. The authors first establish a precise correspondence between the random (unplanted) ensemble and the planted ensemble by introducing the notion of “quiet planting.” In a quiet planting regime the planted solution is statistically indistinguishable from a typical solution of a random instance, meaning that the two ensembles share the same macroscopic properties (e.g., degree distribution, clause density) and therefore can be studied with the same analytical tools.

The central analytical framework combines the cavity method from statistical physics with rigorous reconstruction arguments on trees and first‑ and second‑moment calculations. Using the cavity method, the authors derive replica‑symmetric (RS) and one‑step replica‑symmetry‑breaking (1‑RSB) equations for the model’s free energy and entropy. The reconstruction on trees analysis proceeds by approximating the factor graph of a locked CSP with an infinite‑depth Galton‑Watson tree. By tracking the flow of information from the leaves to the root, they identify a reconstruction threshold at which the root variable’s state can be inferred with non‑vanishing accuracy. This threshold coincides for both the random and planted ensembles, providing a rigorous bridge between them.

First‑moment calculations give the expected number of satisfying assignments, while second‑moment calculations control the variance. When the second moment is of the same order as the square of the first moment, the second‑moment method guarantees that almost all planted instances possess at least one satisfying assignment. Conversely, when the second moment blows up, the method fails, signalling a region where solutions are either absent or extremely hard to find. This leads to the definition of a “hard region” in the planted ensemble, denoted by the interval (