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# Geometric Set Cover and Hitting Sets for Polytopes in R³

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LIPIcs.STACS.2008.1367.pdf
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## Cite As

Sören Lauen. Geometric Set Cover and Hitting Sets for Polytopes in R³. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 479-490, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)
https://doi.org/10.4230/LIPIcs.STACS.2008.1367

## Abstract

Suppose we are given a finite set of points \$P\$ in \$R^3\$ and a collection of polytopes \$mathcal{T}\$ that are all translates of the same polytope \$T\$. We consider two problems in this paper. The first is the set cover problem where we want to select a minimal number of polytopes from the collection \$mathcal{T}\$ such that their union covers all input points \$P\$. The second problem that we consider is finding a hitting set for the set of polytopes \$mathcal{T}\$, that is, we want to select a minimal number of points from the input points \$P\$ such that every given polytope is hit by at least one point. We give the first constant-factor approximation algorithms for both problems. We achieve this by providing an epsilon-net for translates of a polytope in \$R^3\$ of size \$\bigO(frac{1{epsilon)\$.
##### Keywords
• Computational Geometry
• Epsilon-Nets
• Set Cover
• Hitting Sets

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