Maximum dispersion and geometric maximum weight cliques

We consider geometric instances of the problem of finding a set of k vertices in a complete graph with nonnegative edge weights. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. This problem can be considered as a facility location problem, where the objective is to "disperse" a number of facilities, i.e., select a given number of locations from a discrete set of candidates, such that the average distance between selected locations is maximized. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.