A plant location guide for the unsure

This paper studies an extension of the k-median problem where we are given a metric space (V, d) and not just one but m client sets {S _i ⊆ V} ^m _i=1, and the goal is to open k facilities F to minimize: max _i∈[m] {Σ _j∈S _i ^d(j,F)}, i.e., the worst-case cost over all the client sets. This is a "min-max" or "robust" version of the k-median problem; however, note that in contrast to previous papers on robust/stochastic problems, we have only one stage of decision-making-where should we place the facilities? We present an O(log n + log m) approximation for robust k-median: The algorithm is combinatorial and very simple, and is based on reweighting/ Lagrangeanrelaxation ideas. In fact, we give a general framework for (minimization) facility location problems where there is a bound on the number of open facilities. For robust and stochastic versions of such location problems, we show that if the problem satisfies a certain "projection" property, essentially the same algorithm gives a logarithmic approximation ratio in both versions. We use our framework to give the first approximation algorithms for robust/stochastic versions of k-tree, capacitated k-median, and fault-tolerant k-median.