An Empirical Study of Phase Transitions in Binary Constraint Satisfaction Problems

An empirical study of randomly generated binary constraint satisfaction problems reveals that for problems with a given number of variables, domain size, and connectivity there is a critical level of constraint tightness at which a phase transition occurs. At the phase transition, problems change from being soluble to insoluble, and the difficulty of problems increases dramatically. A theory developed by Williams and Hogg [44], and independently developed by Smith [37], predicts where the hardest problems should occur. It is shown that the theory is in close agreement with the empirical results, except when constraint graphs are sparse.