Abstract
In a Stackelberg pricing game a distinguished player, the leader, chooses prices for a set of items, and the other player, the follower, seeks to buy a minimal cost feasible subset of the items. The goal of the leader is to maximize her revenue, which is determined by the sold items and their prices. Typically, the follower’s feasible subsets are given by a combinatorial covering problem. In the Stackelberg shortest path game, for example, the items are edges in a network graph and the follower’s feasible subsets are s-t-paths. This game has been used to model road-toll setting problems by Labbé et al. [14].
We initiate the study of pricing problems where the follower’s feasible subsets are given by a packing problem, e.g., a matching or an independent set problem. We introduce a model that naturally extends packing problems to Stackelberg pricing games. The resulting pricing games have applications related to scheduling.
Our interest is the complexity of computing leader-optimal prices depending on different types of followers. As the main result, we show that the Stackelberg pricing game where the follower is given by the well-known interval scheduling problem is solvable in polynomial time. The interval scheduling problem is equivalent to the independent set problem on interval graphs.
As a complementary result, we prove APX-hardness when the follower is given by the bipartite matching problem. This result also shows APX-hardness for the case where the follower is given by the independent set problem on perfect graphs. On a more general note, we prove \(\varSigma _2^p\)-completeness if the follower is given by a particular packing problem that is NP-complete. In this case, the leader’s pricing problem is hard even if she has an NP-oracle at hand.
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Notes
- 1.
Any interval representation is suitable for our purpose, but we need to fix one.
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Böhnlein, T., Schaudt, O., Schauer, J. (2019). Stackelberg Packing Games. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham. https://doi.org/10.1007/978-3-030-24766-9_18
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