Conditional preference statements have been used to compactly represent preferences over combinatorial domains. They are at the core of CP-nets and their generalizations, and lexicographic preference trees. Several works have addressed the complexity of some queries (optimization, dominance in particular). We extend in this paper some of these results, and study other queries which have not been addressed so far, like equivalence, thereby contributing to a knowledge compilation map for languages based on conditional preference statements. We also introduce a new parameterised family of languages, which enables to balance expressiveness against the complexity of some queries.

This paper considers the task of learning the preferences of users on a combinatorial set of alternatives, as it can be the case for example with online configurators. In many settings, what is available to the learner is a set of positive examples of alternatives that have been selected during past interactions. We propose to learn a model of the users' preferences that ranks previously chosen alternatives as high as possible. In this paper, we study the particular task of learning conditional lexicographic preferences. We present an algorithm to learn several classes of lexicographic preference trees, prove convergence properties of the algorithm, and experiment on both synthetic data and on a real-world bench in the domain of recommendation in interactive configuration.