Local Alterations of the Lagrange Multipliers for Enhancing the Filtering of the AtMostNValue Constraint

Abstract

The reduced cost filtering is a technique that consists in filtering a constraint using the reduced cost of a linear program that encodes this constraint. Sellmann [16] shows that while doing a Lagrangian relaxation of a constraint, suboptimal Lagrange multipliers can provide more filtering than optimal ones. Boudreault and Quimper [5] make an algorithm that locally altered the Lagrange multipliers for the WeightedCircuit constraint to enhance filtering and achieve a speedup of 30%. We seek to design an algorithm like Boudreault and Quimper, but for the AtMostNValue constraint. Based on the work done by Cambazard and Fages [7] on this constraint, we use a subgradient algorithm which takes into consideration the reduced cost to boost the Lagrange multipliers in the optimal filtering direction. We test our methods on the dominating queens and the p-median problem. On the first, we record a speedup of 71% on average. On the second, there are three classes of instances. On the first two, we have an average speedup of 33% and 8%. On the hardest class, we find up to 13 better solutions than the previous algorithm on the 30 instances in the class.