Approximation Schemes for Capacitated Clustering in Doubling Metrics

Abstract: Motivated by applications in redistricting, we consider the uniform capacitated k-median and uniform capacitated k-means problems in bounded doubling metrics. We provide the first QPTAS for both problems and the first PTAS for the uniform capacitated k-median problem for points in R^2. This is the first improvement over the bicriteria QPTAS for capacitated k-median in low-dimensional Euclidean space of Arora, Raghavan, Rao STOC 1998 and arguably the first polynomial-time approximation algorithm for a non-trivial metric.