A plant location guide for the unsure: Approximation algorithms for min-max location problems

This paper studies an extension of the k-median problem under uncertain demand. We are given an n-vertex metric space (V, d) and m client sets $\{S_i\subseteq V\}{}_{i=1}^m$ . The goal is to open a set of k facilities F such that the worst-case connection cost over all the client sets is minimized, i.e., $\underset{F\subseteq V,\mid F\mid =k}{min}\underset{i\in \left[m\right]}{max}\left\{\sum _{j\in S_i}d\right(j,F\left)\right\},$ where for any $F\subseteq V,d(j,F)=\underset{f\in F}{min}d(j,f)$ . This is a ‘min-max’ or ‘robust’ version of the k-median problem. Note that in contrast to the recent papers on robust and stochastic problems, we have only one stage of decision-making where we select a set of k facilities to open. Once a set of open facilities is fixed, each client in the uncertain client-set connects to the closest open facility. We present a simple, combinatorial O(log n + log m)-approximation algorithm for the robust k-median problem that is based on reweighting/Lagrangean-relaxation ideas. In fact, we give a general framework for (minimization) k-facility location problems where there is a bound on the number of open facilities. We show that if the location problem satisfies a certain ‘projection’ property, then both the robust and stochastic versions of the location problem admit approximation algorithms with logarithmic ratios. We use our framework to give the first approximation algorithms for robust and stochastic versions of several location problems such as k-tree, capacitated k-median, and fault-tolerant k-median.