Geometric Set Cover and Hitting Sets for Polytopes in R³

Author Sören Lauen

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Sören Lauen

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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)


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 $\bigO(frac{1{epsilon)$.
  • Computational Geometry
  • Epsilon-Nets
  • Set Cover
  • Hitting Sets


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