On the complexity of the highway problem

In the highway problem, we are given a path, and a set of buyers interested in buying sub-paths of this path; each buyer declares a non-negative budget, which is the maximum amount of money she is willing to pay for that sub-path. The problem is to assign non-negative prices to the edges of the path such that we maximize the profit obtained by selling the edges to the buyers who can afford to buy their sub-paths, where a buyer can afford to buy her sub-path if the sum of prices in the sub-path is at most her budget. In this paper, we show that the highway problem is strongly NP-hard; this settles the complexity of the problem in view of the existence of a polynomial-time approximation scheme, as was recently shown in Grandoni and Rothvoß (2011) [15]. We also consider the coupon model, where we allow some items to be priced below zero to improve the overall profit. We show that allowing negative prices makes the problem APX-hard. As a corollary, we show that the bipartite vertex pricing problem is APX-hard with budgets in {1,2,3}, both in the cases with negative and non-negative prices.