On the Continuous Fermat-Weber Problem

We give the firstexact algorithmic study of facility location problems that deal with finding a median for acontinuum of demand points. In particular, we consider versions of the "continuousk-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 demandregion. 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 theL1 1-median (Fermat-Weber) problem. We also consider the multiple-center version of theL1k-median problem, which we prove is NP-hard for largek.