We investigate a class of location problems where two competing providers place their facilities sequentially and users can decide between the competitors. We assume that both competitors act non-cooperatively and aim at maximizing their own benefit. We investigate the complexity and approximability of such problems on graphs, in particular on simple graph classes such as trees and paths. We also develop fast algorithms for single competitive location problems where each provider places one single facilty.
A location problem aims at finding suitable locations for new facilities that are to be opened. Given a set of potential locations, its quality is measured by the distances to the customers of the facilities. Prominent examples are the k-median and the k-center problem. Often, facilities and customers are represented by nodes of an edge-weighted graph. Distances are given by the lengths of shortest paths.
Many location problems dealt with in the literature assume the existence of a single monopolistic provider who wants to open a number of new facilities and looks for a set of good locations. In contrast, competitive location investigates scenarios where two or more competing providers place their facilities and customers can decide between the providers.
We consider models with two sequentially acting competitors, leader and follower. We assume that both competitors offer the same type of good or service at the same price. Hence the user preference can be expressed solely in terms of distances to the locations of the facilities. Every customer chooses the closest competitor. Once the leader has chosen a location, it is the follower's turn to determine a location maximizing his own revenue (the total demand of his customers). Hence the follower's reaction is predictable, which the leader can take into account when making the initial decision. We assume that the competitors act non-cooperatively.
The complexity status of the leader problem on tree graphs has been a long-standing open question (Hakimi, 1990). One of our main results is that the leader problem is NP-hard even on paths thereby answering this question. (For more detailed information we refer to the journal article.) On the positive side we give a fully polynomial-time approximation scheme for paths.
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