Reducing traffic congestion via toll pricing has been a central topic in the operations research and transportation literature and, recently, it has been implemented in several cities all over the world. Since, in practice, it is not feasible to impose tolls on every edge of a given traffic network, we study the resulting mathematical problem of computing tolls on a predefined subset of edges of the network so as to minimize the total travel time of the induced equilibrium flow. We first present an analytical study for the special case of parallel edge networks highlighting the intrinsic complexity and nonconvexity of the resulting optimization problem. We then present algorithms for general networks for which we systematically test the solution quality for large-scale network instances. Finally, we discuss the related optimization problem of computing tolls subject to a cardinality constraint on the number of edges that have tolls.