On Metric Clustering to Minimize the Sum of Radii

Authors:
Matt Gibson;Gaurav Kanade;Erik Krohn;Imran A. Pirwani;Kasturi Varadarajan
Affiliations:
Department of Computer Science, University of Iowa, Iowa City, USA IA 52242-1419;Department of Computer Science, University of Iowa, Iowa City, USA IA 52242-1419;Department of Computer Science, University of Iowa, Iowa City, USA IA 52242-1419;Department of Computer Science, University of Iowa, Iowa City, USA IA 52242-1419;Department of Computer Science, University of Iowa, Iowa City, USA IA 52242-1419
Venue:
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
Year:
2008

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Abstract

Given an n-point metric (P,d) and an integer k 0, we consider the problem of covering Pby kballs so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in nO(logn·logΔ)time and returns with high probability the optimal solution. Here, Δis 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.