On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the tollbooth problem, we are given a tree T = (V,E) with n edges, and a set of m customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and is willing to pay at most her budget to purchase her path. The objective of the problem is to price the edges of T such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the highway problem, is when T is restricted to be a path. In this paper, we study the tollbooth and highway problems. For the tollbooth problem, we present a randomized O(log n)-approximation, improving on the current best O(logm)-approximation, since n can be assumed to be at most 3m. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of T. In this case, we present an algorithm that returns a (1−ǫ)-approximation, for any ǫ > 0 and runs in quasi-polynomial time. Finally, we show that the highway problem that was earlier only known to be weakly NP-hard, is in fact strongly NP-hard.