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Covering many or few points with unit disks. (English) Zbl 1187.68716

Summary: 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 \(m\geq 1\), such that the total weight of the points from \(P\) inside the union of the disks is maximized. We present algorithms that compute, for any fixed \(\varepsilon >0\), a \((1 - \varepsilon )\)-approximation to the optimal solution in \(O(n \log n)\) 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 computes, for any fixed \(\varepsilon >0\), in \(O(n \log ^{2} n)\) expected time a disk that is, with high probability, a \((1+\varepsilon )\)-approximation to the optimal solution.

MSC:

68W25 Approximation algorithms
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