Counter constraints are a natural representation of constraints on the finite capacity of resources in resource-allocation type problems. They are a generic family of non-binary constraints that limit the number of variables that may be assigned particular values. Counter constraints can be represented by binary constraints, at a cost. We analyse the cost, show how a counter can be represented as a linear number of binary constraints, and demonstrate empirically that even with the optimal reduction, an explicit representation of counters is preferable to their representation as a set of binary constraints. For counter constraints, value ordering is essential. An heuristic for value ordering on constraint satisfaction problems (CSP), based on the estimated likelihood of a solution, is presented. The proposed value ordering heuristic is useful for counter constraints, as well as for binary CSPs, where it can be used to approximate the number of solutions consistent with a particular value assignment to a variable. The proposed value ordering heuristic integrates counter constraints with binary constraint networks in a novel manner. Counter constraints are problematic for most heuristics, which are local in scope, yet we demonstrated empirically that the proposed value ordering heuristic is significantly superior to heuristics used in previous work.
