On the Continuous Fermat-Weber Problem

Abstract

We give the first exact algorithmic study of facility location problems that deal with finding a median for a continuum of demand points. In particular, we consider versions of the "continuous k-median (Fermat-Weber) problem" where the goal is to select one or more center points that minimize the average distance to a set of points in a demand region. In such problems, the average is computed as an integral over the relevant region, versus the usual discrete sum of distances. The resulting facility location problems are inherently geometric, requiring analysis techniques of computational geometry. We provide polynomial-time algorithms for various versions of the L 1 1-median (Fermat-Weber) problem. We also consider the multiple-center version of the L 1 k-median problem, which we prove is NP-hard for large k. MSC Classification: 90B85, 68U05ACM Classification: F.2.2

Finding a Hausdorff Core of a Polygon: On Convex Polygon Containment with Bounded Hausdorff Distance

“…That suggests our problem may be equally poorly-behaved. Fekete et al [10] considered the continuous Weber problem under the 1 distance metric.…”

Vehicle placement to intercept moving targets

“…Now, given a convex region Q ⊂ R and a density function ψ : Q → R ≥0 , the median (cf. Fekete et al (2005)) p med is the unique global minimizer of…”

“…Megiddo and Supowit (1984) and Zemel (1984)). For a single vehicle, the average distance to a random point, generated according to a probability density function is given by the Weber or the continuous 1-median function, for which there exists a global minimizer as shown in Fekete et al (2005), termed as the median. For multiple distinct vehicle locations, the expected distance between a random point generated according to a probability density and one of the locations is known in literature as the continuous Weber or the continuous multi-median function, e.g., see Drezner (1995).…”

A polynomial-time algorithm for the discrete facility location problem with limited distances and capacity constraints

“…The minisum problem, also known as the Weber problem (cf. Wesolowsky, 1993;Fekete et al, 2005) is a central problem in location theory. It refers to a situation in which there exists a set of demand points and the location of a facility must be chosen such that the total sum of the weighted distances from the points to the facility is minimized.…”