Geometric Set Cover and Hitting Sets for Polytopes in R³

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