Weighted Constrained CP-nets: an Extension of Constrained CP-nets with Weighted Constraints

A Conditional Preference Network (CP-net) is a graphical model widely used to represent qualitative preferences in many real-world applications. Preference elicitation, representation, and reasoning plays an essential role in e-commerce and other applications relying on users’ preferences and desires. However, managing preferences often comes with dealing with hard and soft constraints. While hard constraints are requirements that can either be satisfied or violated, this two-level of satisfiability can be generalized to multiple levels through soft constraints. In this context, our main objective is to manage conditional and qualitative preferences together with hard and soft constraints, within a unique model. The constrained CP-net graphical model has been proposed to manage both conditional preferences and hard constraints. In order to include soft constraints, we extend the underlying constraint network of the constrained CP-net to a Weighted Constraint Satisfaction Problem (WCSP). A WCSP is a CSP where (soft) constraints can be violated or satisfied with associated costs. More precisely, a cost function is associated to each constraint. We call the Weighted Constrained CP-nets (WCCP-net) the new model we propose. Like for CP-nets and Constrained CP-nets, there are two queries to consider for WCCP-nets: outcome optimization and outcome comparison. The outcome optimization query in the case of a constrained CP-net consists in finding the set of feasible solutions that are not dominated by any other solutions. This set is called the Pareto optimal set. In the case of the WCCP-net, this task consists of finding those solutions, from the Pareto optimal set, that maximize the total cost function of the underlying WCSP. This is a NP-hard problem that we tackle using a variant of the branch and bound algorithm that has been proposed to solve the outcome optimization in the case of constrained CP-nets.