Dynamic sum-radii clustering

Abstract

Real networks have in common that they evolve over time and their dynamics have a huge impact on their structure. Clustering is an efficient tool to reduce the complexity to allow representation of the data. In 2014, Eisenstat et al. introduced a dynamic version of this classic problem where the distances evolve with time and where coherence over time is enforced by introducing a cost for clients to change their assigned facility. They designed a Θ(ln n)-approximation. An O(1)- approximation for the metric case was proposed later on by An et al. (2015). Both articles aimed at minimizing the sum of all client-facility distances; however, other metrics may be more relevant. In this article we aim to minimize the sum of the radii of the clusters instead. We obtain an asymptotically optimal Θ(ln n)-approximation algorithm where n is the number of clients and show that existing algorithms from An et al. (2015) do not achieve a constant approximation in the metric variant of this setting.