Reverse facility location problems

In the Nearest Neighbor problem (NN), the objects in the database that are nearer to a given query object than any other objects in the database have to be found. In the conceptually inverse problem, Reverse Nearest Neighbor problem (RNN), objects that have the query object as their nearest neighbor have to be found. Reverse Nearest Neighbors queries have emerged as an important class of queries for spatial and other types of databases. The concept has been introduced by Korn et al. [10, 11], where a large number of applications in marketing and decision support system are given. See [17] for a recent survey on the current state-of-art and open geometric problems in another application area. The RNN problem itself has several variants, namely, the monochromatic, bichromatic, static or dynamic versions. In the monochromatic case, all points have the same color. In the bichromatic case, the point set consists of both red and blue points and the problem is that given a query point of one color, one needs to find the points of the other color for which the query point is a bichromatic nearest neighbor. In the static version of the problem, the distances between the points in the set remain unchanged whereas in the dynamic problem they may change. Some related work includes [4, 12, 14, 16]. This paper considers the static bichromatic variant in which the data points are of two categories. In particular, we define RNN facility location problems, in which some points are designated as facilities and others as customers. In this setting, a reverse nearest neighbor query asks for the set of customers affected by the opening of a new facility at some point, assuming all customers go to their nearest facility. We study optimization problems that arise when considering various optimization criteria: maximizing the number of poten-