Worst-case and Probabilistic Analysis of a Geometric Location Problem

Abstract

We consider the problem of choosing K "medians" among n points on the Euclidean plane such that the sum of the distances from each of the n points to its closest median is minimized. We show that this problem is NP-complete. We also present two heuristics that produce arbitrarily good solutions with probability going to 1. One is a partition heuristic, and works when K grows lineraly -- or almost so -- with n. The other is the "honeycomb" heuristic, and is applicable to rates of grother of K of the form K ~ n^Є, 0<Є<1.