The solution space structure of planted constraint satisfaction problems with growing domains

Abstract

Planting a solution into the random revised B (RB) model, which is a prototype of the random constraint satisfaction problem (CSP) with growing domains, can generate very hard satisfiable CSP benchmarks. We study the solution space structure of the planted RB model. In the thermodynamic limit, by the first moment method we find that this model goes through four phase transitions as the constraint density increases. In the replica symmetric phase, what we call the independent phase transition occurs, after which the planted cluster (cluster containing the planted solution) is separated from the giant cluster. From then on, the solution space except the planted cluster goes through the same clustering phase transition and the same satisfiability phase transition as on the random RB model. The planted cluster goes through the isolated phase transition, after which the planted cluster contains only one solution. When the instances have about 102 variables, experiments show that the last three phase transitions have already appeared as in the thermodynamic limit. This phase diagram provides strong evidence that this model can generate very hard satisfiable CSP benchmarks. Finally, we find when the constraint density goes to infinity, there is a single smooth valley in the energy landscape.