Maximum dispersion and geometric maximum weight cliques

TL;DR

This paper studies a facility dispersion problem aiming to maximize average distances between selected locations, providing optimal algorithms for fixed k and approximation schemes for variable k in d-dimensional space.

Contribution

It introduces new algorithms for dispersing facilities, including a linear-time optimal solution for fixed k and a polynomial-time approximation scheme for variable k.

Findings

Linear-time algorithm for fixed k optimal solution

Polynomial-time approximation scheme for variable k

Improved performance over previous 2-approximation algorithms

Abstract

We consider a facility location problem, where the objective is to ``disperse'' a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. 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. 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.