Worst-Case and Probabilistic Analysis of a Geometric Location Problem

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 linearly—or almost so—with n. The other is the “honeycomb” heuristic, and is applicable to rates of growth of K of the form K n^, 0 < < 1.