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 et al., Networks, 46 (2005), 57-67) proved that this problem is NP-hard, and gave an O(log m)-approximation algorithm, where m denote the number of arcs to be priced. In this note, we show that the problem is also APX-hard. (C) 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 117-120 2011