The Constrainedness of Arc Consistency

We show that the same methodology used to study phase transition behavior in NP-complete problems works with a polynomial problem class, establishing arc consistency in constraint satisfaction problems. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NP-complete problems, predicts the location of a phase transition in establishing arc consistency. Finite size scaling models both the scaling of this transition and of the computational cost for the algorithmAC3. On problems at the phase transition, this model of computational cost agrees with the theoretical worst case for AC3. As with NP-complete problems, constrainedness can be used as the basis for a heuristic. In this case, it gives a constraint ordering heuristic that can reduce the number of checks performed by AC3.