Abstract
Given a metric (V,d) and an integer k, we consider the problem of covering the points of V with at most k clusters so as to minimize the sum of radii or the sum of diameters of these clusters. The former problem is called the Minimum Sum Radii (MSR) problem and the latter is the Minimum Sum Diameters (MSD) problem. The current best polynomial time algorithms for these problems have approximation ratios 3.504 and 7.008, respectively [2]. For the MSR problem, we give an exact algorithm when the metric is the shortest-path metric of an unweighted graph and there cannot be any singleton clusters. For the MSD problem on the plane with Euclidean distances, we present a polynomial time approximation scheme.
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References
Capoyleas, V., Rote, G., Woeginger, G.: Geometric clusterings. Journal of Algorithms 12(2), 341–356 (1991)
Charikar, M., Panigrahy, R.: Clustering to minimize the sum of cluster diameters. Journal of Computer and System Sciences 68(2), 417–441 (2004)
Doddi, S., Marathe, M.V., Ravi, S.S., Taylor, D.S., Widmayer, P.: Approximation algorithms for clustering to minimize the sum of diameters. Nordic J. of Computing 7(3), 185–203 (2000)
Gibson, M., Kanade, G., Krohn, E., Pirwani, I.A., Varadarajan, K.: On metric clustering to minimize the sum of radii. Algorithmica (2010)
Gibson, M., Kanade, G., Krohn, E., Pirwani, I.A., Varadarajan, K.: On clustering to minimize the sum of radii. SIAM Journal on Computing 41(1), 47–60 (2012)
Hansen, P., Jaumard, B.: Minimum sum of diameters clustering. Journal of Classification 4(2), 215–226 (1987)
Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10(2), 180–184 (1985)
Jung, H.W.: Über die kleinste kugel, die eine räumliche figur einschliesst. J. Reine Angew. Math. 123, 241–257 (1901)
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© 2012 Springer-Verlag Berlin Heidelberg
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Behsaz, B., Salavatipour, M.R. (2012). On Minimum Sum of Radii and Diameters Clustering. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_7
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DOI: https://doi.org/10.1007/978-3-642-31155-0_7
Publisher Name: Springer, Berlin, Heidelberg
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