Random constraint satisfaction: A more accurate picture

Recently there has been a great amount of interest in Random Constraint Satisfaction Problems, both from an experimental and a theoretical point of view. Rather intriguingly, experimental results with various models for generating random CSP instances suggest a “threshold-like” behavior and some theoretical work has been done in analyzing these models when the number of variables becomes large (asymptotic). In this paper we prove that the models commonly used for generating random CSP instances do not have an asymptotic threshold. In particular, we prove that as the number of variables becomes large, almost all instances they generate are trivially overconstrained. We then present a new model for random CSP and, in the spirit of random k-SAT, we derive lower and upper bounds for its parameters so that instances are "almost surely" underconstrained and overconstrained, respectively. Finally, for the case of one of the popular models in Artificial Intelligence we derive sharper estimates for the probability of being overconstralned, as a function of the number of variables.