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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References:

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