Profit-maximizing pricing for tollbooths

The input to the tollbooth problem is a graph G = (V,E) and a set of m buyers Pi where each buyer is interested in buying a path Pi connecting si, ti ∈ V . Each buyer comes with a budget b(Pi), a positive real number. The problem is to set non-negative prices to the edges E of G. A buyer buys her path Pi if the sum of prices on Pi is at most her budget. If a buyer buys her path, we obtain a profit equal to the sum of prices on the chosen path, and otherwise we get nothing from the buyer. The objective of the problem is to find prices for the edges that maximizes the revenue obtained by selling the edges to the buyers who can afford their path. There is no restriction on the number of buyers that can simultaneously use an edge. In this paper, we consider the tollboth problem on a tree. Hence, each si, ti path is uniquely determined. A upward instance is when the tree is rooted at a vertex r, and ti is an ancestor of si for each buyer Pi, i = 1, . . . ,m. We show that any solution to a naturally defined upward instance obtained from a tollbooth instance can be converted back to a solution to the orginial problem at a loss of at most a constant factor in the approximation. Using known results on the tollbooth problem for instances, we obtain improved algorithms for the tollbooth problem on trees.