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Covering many or few points with unit disks
Berg, Mark de ; Cabello, Sergio ; Har-Peled, Sariel
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 ▫$m \ge 1$▫, 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 ▫$\varepsilon > 0$▫ 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.
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