Polynomial cases of the tarification problem

We consider the problem of determining a set of optimal tariffs for an agent in a network, who owns a subset of the arcs of the network, and who wishes to maximize his revenues on this subset from a set of clients that make use of the network.The general variant of this problem is NP-hard, already with a single client. This paper introduces several new polynomially solvable special cases. An important case is the following.For multiple clients, if the number of tariff arcs is bounded from above, we can solve the problem by a polynomial number of linear programs (each of which is of polynomial size). Furthermore, we show that the parametric tarification problem and the single arc fixed charge tarification problem can be solved in polynomial time.