Computational Aspects of a 2-Player Stackelberg Shortest Paths Tree Game

• Authors: Davide Bilò;Luciano Gualà;Guido Proietti;Peter Widmayer
• Venue: WINE '08 Proceedings of the 4th International Workshop on Internet and Network Economics
• Year: 2008

Let a communication network be modelled by a directed graphG = (V,E) of n nodes and m edges. We consider aone-round two-player network pricing game, the Stackelberg ShortestPaths Tree (StackSPT) game. This is played on G, by assuming thatedges in E are partitioned into two sets: a set E F of edges with afixed positive real weight, and a set E P of edges that should bepriced by one of the two players (the leader). Given adistinguished node r ∈ V, the StackSPT game isthen as follows: the leader prices the edges in E P in such a waythat he will maximize his revenue, knowing that the other player(the follower) will build a shortest paths tree of G rooted at r,say S(r), by running a publicly available algorithm. Quitenaturally, for each edge selected in the solution, theleader’s revenue is assumed to be equal to the loaded priceof an edge, namely the product of the edge price times the numberof paths from r in S(r) that use it. First, we show that theproblem of maximizing the leader’s revenue is NP-hard as soonas |E P | = Θ(n). Then, in search of aneffective method for solving the problem when the size of E P isconstant, we focus on the basic case in which |E P| = 2, and we provide an efficient O(n 2 logn) timealgorithm. Afterwards, we generalize the approach to the case |E P| = k, and we show that it can be solved inpolynomial time whenever k = O(1).