Abstract
Let P be a set of n weighted points. We study approximation algorithms for the following two continuous facility-location problems.
In the first problem we want to place m unit disks, for a given constant m≥1, such that the total weight of the points from P inside the union of the disks is maximized. We present a deterministic algorithm that can compute, for any ε>0, a (1−ε)-approximation to the optimal solution in O(n logn + ε \(^{{\rm -4}{\it m}}\)log\(^{\rm 2{\it m}}\) (1/ε)) time.
In the second problem we want to place a single disk with center in a given constant-complexity region X such that the total weight of the points from P inside the disk is minimized. Here we present an algorithm that can compute, for any ε>0, with high probability a (1+ε)-approximation to the optimal solution in O(n (log3 n + ε − 4 log2 n )) expected time.
MdB was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301. SC was partially supported by the European Community Sixth Framework Programme under a Marie Curie Intra-European Fellowship, and by the Slovenian Research Agency, project J1-7218.
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de Berg, M., Cabello, S., Har-Peled, S. (2007). Covering Many or Few Points with Unit Disks. In: Erlebach, T., Kaklamanis, C. (eds) Approximation and Online Algorithms. WAOA 2006. Lecture Notes in Computer Science, vol 4368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11970125_5
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DOI: https://doi.org/10.1007/11970125_5
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