Stackelberg Shortest Path Tree Game, Revisited

Let G(V,E) be a directed graph with n vertices and m edges. The edges E of G are divided into two types: EF and EP . Each edge of EF has a fixed price. The edges of EP are the priceable edges and their price is not fixed a priori. Let r be a vertex of G. For an assignment of prices to the edges of EP , the revenue is given by the following procedure: select a shortest path tree T from r with respect to the prices (a tree of cheapest paths); the revenue is the sum, over all priceable edges e, of the product of the price of e and the number of vertices below e in T . Assuming that k = |EP | ≥ 2 is a constant, we provide a data structure whose construction takes O(m+n logk−1 n) time and with the property that, when we assign prices to the edges of EP , the revenue can be computed in (log k−1 n). Using our data structure, we save almost a linear factor when computing the optimal strategy in the Stackelberg shortest paths tree game of [D. Bilò and L. Gualà and G. Proietti and P. Widmayer. Computational aspects of a 2-Player Stackelberg shortest paths tree game. Proc. WINE 2008].