Stackelberg network pricing is hard to approximate

In the Stackelberg network pricing problem, one has to assign tariffs to a certain subset of the arcs of a given transportation network. The aim is to maximize the amount paid by the user of the network, knowing that the user will take a shortest st-path once the tariffs are fixed. Roch, Savard, and Marcotte (Networks, Vol. 46(1), 57–67, 2005) proved that this problem is NP-hard, and gave an O(logm)-approximation algorithm, where m denote the number of arcs to be priced. In this note, we show that the problem is also APX-hard.