Learning solution preferences in constraint problems

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Abstract
Usually, not all the solutions of a finite domain constraint satisfaction problem (CSP) are equally desirable: some of them may be preferred to others. However, classical CSPs do not allow for this more informative kind of knowledge representation. On the other hand, semiring-based CSPs (SCSPs), where a value is associated with each tuple in each constraint, generate solutions with a corresponding value attached that can be interpreted as the level of preference of that solution. Sometimes, however, even standard SCSPs are not enough, since one may know preferences over some of the solutions but have no idea on how to code this knowledge into the SCSP. In this paper we consider this situationand propose to address it by first defining a classical CSP and giving some examples of solution preferences, and then learning the corresponding SCSP that behaves as the initial CSP (that is, it has the same solutions) and matches the preferences specified in the examples. In other words, we use the examples as the training set, and we employ a learning scheme to adjust the values to be attached to the constraint tuples, such that the resulting solution preferences coincide with the examples. In this way, we make the SCSP framework more flexible, since it can be used also when it is difficult to assign values to tuples and instead it is easier to rate some of the solutions.