Preference reasoning is a central problem in decision support. There exist various ways to interpret a set of qualitative preferences. Conditional preference logics allow to deal with semantics such as optimistic, pessimistic, strong or not. In this paper, we study the complexity of the main problems in optimistic/pessimistic preference logic: undominated, consistency and dominance. We show that they are all NP-hard in general, with some becoming polynomial under specific semantics. Our second contribution is to show that the dominance problem, which has an online component in its definition, is compilable to polynomial time.