Ensuring truthfulness amongst self-interested agents bidding against one another in an auction can be computationally expensive when prices are determined using the Vickrey-Clarke-Groves (VCG) mechanism. This mechanism guarantees that each agent's dominant strategy is to tell the truth, but it requires solving n + 1 optimization problems when the overall optimal solution involves n agents. This paper first examines a case-study example demonstrating how Operations Research techniques can be used to compute Vickrey prices efficiently. In particular, the case-study focuses on the Assignment Problem. We show how, in this case, Vickrey prices can be computed in the same asymptotic time complexity as that of the original optimization problem. This case-study can be seen as serving a pedagogical role in the paper illustrating how Operations Research techniques can be used for fast Vickrey pricing. We then propose a Constraint Programming approach that can be used in a more general context, where nothing is assumed about the nature of the constraints that must be satisfied or the structure of the underlying problem. In particular, we demonstrate how nogood learning can be used to improve the efficiency of constraint-based Vickrey pricing in combinatorial auctions.