Finding the next solution in constraint-and preference-based knowledge representation formalisms

In constraint or preference reasoning, a typical task is to compute a solution, or an optimal solution. However, when one has already a solution, it may be important to produce the next solution following the given one in a linearization of the solution ordering where more preferred solutions are ordered first. In this paper, we study the computational complexity of finding the next solution in some common preference-based representation formalisms. We show that this problem is hard in general CSPs, but it can be easy in tree-shaped CSPs and tree-shaped fuzzy CSPs. However, it is difficult in weighted CSPs, even if we restrict the shape of the constraint graph. We also consider CP-nets, showing that the problem is easy in acyclic CP-nets, as well as in constrained acyclic CP-nets where the (soft) constraints are tree-shaped and topologically compatible with the CP-net.