On the Complexity of Stackelberg Matroid Pricing Problems

In a Stackelberg pricing problem a distinguished player, the leader, chooses prices for a set of items, and one or several other players, the followers, seeks to buy a feasible subset of the items with minimal costs. The leader’s goal is to maximize her revenue, which is determined by the sold items and their prices. We are interested in cases where the followers’ feasible subsets are given by a combinatorial optimization problem. For example, a pricing problem based on the shortest path problem was used by Labbé et al. [15] to model road-toll setting scenarios. In this paper, we consider Stackelberg pricing problems that are based on matroids. The followers seek to buy a subset that is a basis. More specifically, we consider uniform, partition and laminar matroids. We study the complexity of computing leader-optimal prices for a single and multiple followers. We show that optimal prices can be computed in polynomial time for all three matroids if there is one follower. In general, such pricing problems based on matroids are APX-hard (see [11]). For multiple followers, we show that computing optimal prices for uniform matroids can be done in polynomial time. However, for partition and laminar matroids the pricing problem becomes NP-hard.