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

Sören Laue (1)

1 MPII - Max-Planck-Institut für Informatik

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

Mots clés: Computational Geometry; Epsilon-Nets; Set Cover; Hitting Sets

Domaines: Géométrie algorithmique [cs.CG]

Sören Laue. Geometric Set Cover and Hitting Sets for Polytopes in $R^3$. STACS 2008, Feb 2008, Bordeaux, France. pp.479-490. ⟨hal-00232968⟩