On the continuous Weber and k-median problems

Sándor P. Fekete, Joseph S. B. Mitchell, and Karin Weinbrecht

Extended abstract in: 16th Annual Symposium on Computational Geometry (SoCG 2000), 70-79

location theory, Weber problem, k-median, median, continuous demand, computational geometry, geometric optimization, shortest paths, rectilinear norm, computational complexity

We give the first exact algorithmic study of facility location problems having a continuum of demand points. In particular, we consider versions of the "continuous k-median (Weber) problem" where the goal is to select one or more center points that minimize 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 L₁ 1-median (Weber) problem. We also consider the multiple-center version of the L₁ k-median problem, which we prove is NP-hard for large k.

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