Geometric Set Cover and Hitting Sets for Polytopes in R3

Suppose we are given a finite set of points P in R 3 and a collection of polytopes 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 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 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 O(1 ǫ).