Algorithmica (2010) 57: 484–498 DOI 10.1007/s00453-009-9282-7 On Metric Clustering to Minimize the Sum of Radii Matt Gibson · Gaurav Kanade · Erik Krohn · Imran A. Pirwani · Kasturi Varadarajan Received: 22 August 2008 / Accepted: 19 January 2009 / Published online: 12 February 2009 © Springer Science+Business Media, LLC 2009 Abstract Given an n-point metric (P,d) and an integer k> 0, we consider the prob- lem of covering P by k balls so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in n O(log n·log ) time and returns with high probability the optimal solution. Here,  is the ratio between the maximum and minimum interpoint distances in the metric space. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and in metrics of constant doubling dimension. Keywords Clustering · Polynomial time · Approximation algorithm Work of M. Gibson, G. Kanade, E. Krohn, and K. Varadarajan was partially supported by NSF CAREER award CCR 0237431. Work of I.A. Pirwani was partially supported by Alberta Ingenuity. Most of this work was done while I.A. Pirwani was at the University of Iowa, Iowa City, IA 52242, USA. Part of this work was done while K. Varadarajan was visiting the Institute for Mathematical Sciences, Chennai, India. M. Gibson · G. Kanade · E. Krohn · K. Varadarajan ( ) Department of Computer Science, University of Iowa, Iowa City, IA 52242-1419, USA e-mail: kvaradar@cs.uiowa.edu M. Gibson e-mail: mrgibson@cs.uiowa.edu G. Kanade e-mail: gkanade@cs.uiowa.edu E. Krohn e-mail: eakrohn@cs.uiowa.edu I.A. Pirwani Department of Computing Science, University of Alberta, Edmonton, Alberta T6G 2E8, Canada e-mail: pirwani@cs.ualberta.ca