Geometric Set Cover and Hitting Sets for Polytopes in $R^3$

Details Geometric Set Cover and Hitting Sets for Polytopes in $R^3$ Geometric Set Cover and Hitting Sets for Polytopes in $R^3$

2008-02-20

arxiv.org/abs/0802.2861v1

Dans Proceedings of the 25th Annual Symposium on the Theoretical Aspects of Computer Science - STACS 2008, Bordeaux : France (

Computer Science

Computational Geometry

Scientific paper

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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