Discrepancy-based additive bounding for the AllDifferent constraint

In this paper we show how to exploit in Constraint Programming (CP) a well-known integer programming technique, the additive bounding procedure, when using Limited Discrepancy Search (LDS). LDS is an effective search strategy based on the concept of discrepancy, i.e., a branching decision which does not follow the suggestion of a given heuristic. The property of a node to have an associated discrepancy k can be modeled (and enforced) through a constraint, called k-discrepancy constraint. Our key result is the exploitation of the k-discrepancy constraint to improve the bound given by any relaxation of a combinatorial optimization problem by using the additive bounding idea. We believe that this simple idea can be effectively exploited to tighten relaxations in CP solvers and speed up the proof of optimality. The general use of additive bounding in conjunction with LDS has been presented in [14]. Here we focus on a particular case where the AllDifferent constraint is part of the CP model. In this case, the integration of additive bound in CP is particularly effective.