Branch-and-Bound for Soft Constraints Based on Partially Ordered Degrees of Preference

The handling of partially ordered degrees of preference can be important for constraint-based languages, especially when there is more than one criterion that we are interested in optimising. This paper describes branch and bound algorithms for a very general class of soft constraint optimisation problems. At each node of the search tree, a propagation mechanism is applied which generates an upper bound (since we are maximising) of the preference degrees of all complete assignments below the node. We show how this propagation can be achieved using an extended mini-buckets algorithm. However, since the degrees of preference ordering are only partially ordered, such an upper bound can be very uninformative, and so it can be desirable to instead generate an upper bound set, which contains an upper bound for the degree of preference for each complete assignment below the node. It is shown how such propagation can also be achieved using this extended mini-buckets approach.