Approximation algorithms for hierarchical location problems

We formulate and (approximately) solve hierarchical versions of two prototypical problems in discrete location theory, namely, the metric uncapacitated k-median and facility location problems. Our work yields new insights into hierarchical clustering, a widely used technique in data analysis. For example, we show that every metric space admits a hierarchical clustering that is within a constant factor of optimal at every level of granularity with respect to the average (squared) distance objective. A key building block of our hierarchical facility location algorithm is a constant-factor approximation algorithm for an “incremental” variant of the facility location problem; the latter algorithm may be of independent interest.