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Beschreibung:
Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks.