Analysis Of Reduced Costs Filtering For Alldifferent And Minimumweight Alldifferent Global Constraints

An incomplete filtering technique known as variable fixing has been used in integer programming for a long time. It relies on the reduced costs of the variables given by an optimal dual solution of the linear relaxation. Reduced-costs are used to detect some of the 0/1 variables that must be fixed to either 0 or 1 in any solution improving the best known. Reduced cost based filtering was introduced in CP for a global constraint referred to as MINIMUM WEIGHT ALLDIFFERENT and to the best of our knowledge, no analysis of this filtering technique has ever been performed. We therefore propose an analysis of reduced costs filtering for this constraint, showing that arc-consistency can be achieved with reduced-costs of n dual solutions and that this bound is sharp. For ALLDIFFERENT, a single dual solution is enough. From a practical side, our end goal is the design of incomplete but anytime primal-dual filtering approaches. We illustrate this idea on the MINIMUM WEIGHT ALLDIFFERENT where a near-complete filtering can be done in shorter times.