To efficiently solve exact discrete optimization problems, branch and bound algorithms require tight bounds. In constraint programming, for optimization, soft arc consistencies typically derive much stronger bounds than those offered by domain or bound consistencies applied to a cost variable. The reason is that soft local consistencies exchange marginal cost information between variables whereas domain consistencies rely only on shrinking domains, which is less informative. However, CP solvers equipped with soft arc consistencies have so far offered limited support for efficient processing of global constraints. In this work, we show how we can efficiently enforce soft local consistency over the AllDifferent constraint, relying on algorithms for the Linear Assignment Problem (LAP). We implement this propagator in toulbar2, the state-of-the-art weighted CP solver exploiting soft local consistencies for bounding. We show that, equipped with this new propagator, toulbar2 outperforms state-of-the-art domain consistency-based CP as well as integer programming solvers for the Quadratic Assignment Problem and shows better performance for miniCOP instances of the 2024 XCSP competition with AllDifferent constraints.

In Constraint Programming, solving discrete minimization problems with hard and soft constraints can be done either using (i) soft global constraints, (ii) a reformulation into a linear program, or (iii) a reformulation into local cost functions. Approach (i) benefits from a vast catalog of constraints. Each soft constraint propagator communicates with other soft constraints only through the variable domains, resulting in weak lower bounds. Conversely, the approach (ii) provides a global view with strong bounds, but the size of the reformulation can be problematic. We focus on approach (iii) in which soft arc consistency (SAC) algorithms produce bounds of intermediate quality. Recently, the introduction of linear constraints as local cost functions increases their modeling expressiveness. We adapt an existing SAC algorithm to handle linear constraints. We show that our algorithm significantly improves the lower bounds compared to the original algorithm on several benchmarks, reducing solving time in some cases.