Abstract
Given an n-point metric (P,d) and an integer k>0, we consider the problem 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 Δ) time and returns with high probability the optimal solution. Here, Δ 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.
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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.
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Gibson, M., Kanade, G., Krohn, E. et al. On Metric Clustering to Minimize the Sum of Radii. Algorithmica 57, 484–498 (2010). https://doi.org/10.1007/s00453-009-9282-7
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DOI: https://doi.org/10.1007/s00453-009-9282-7