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DOI: 10.4230/LIPIcs.STACS.2008.1367

Geometric Set Cover and Hitting Sets for Polytopes in R³

Authors: Sören Lauen

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

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 $\bigO(frac{1{epsilon)$.

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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)

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@InProceedings{lauen:LIPIcs.STACS.2008.1367,
  author =	{Lauen, S\"{o}ren},
  title =	{{Geometric Set Cover and Hitting Sets for Polytopes in R³}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{479--490},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1367},
  URN =		{urn:nbn:de:0030-drops-13675},
  doi =		{10.4230/LIPIcs.STACS.2008.1367},
  annote =	{Keywords: Computational Geometry, Epsilon-Nets, Set Cover, Hitting Sets}
}

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Geometric Set Cover and Hitting Sets for Polytopes in R³

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