The Phase Transition Behaviour of Non-binary Forward Checking Algorithms

In this paper the phase transition, that is to say the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solution as the constraints become tighter, is investigated by examining n-ary Constraint Satisfaction Problems (CSPs). The paper reports a series of experiments with samples of randomly-generated problems, using a family of non-binary Forward Checking algorithms. We observe the behaviour of these algorithms with respect to the phase transition properties of n-ary Constraint Satisfaction Problems (CSPs). An n-ary random problem has an arity for the constraints, a number of variables, a uniform domain size, connectivity, and tightness of constraints. It was observed that for a problem with a given arity, number of variables, domain size, and connectivity there is a critical level of constraint tightness at which a phase transition occurs, such that the problems change from being soluble to insoluble. At the phase transition the search effort becomes orders of magnitude harder than elsewhere. It is shown that the location of the really hard problems can be predicted with a reasonable degree of accuracy. The accuracy of prediction based on the expected number of solutions is discussed. MOTS-CLÉS : Transition de phase, satisfaction de contraintes