Approximation Algorithms for Geometric Optimization Problems

Suppose we are given a finite set of points P in R^3 and a collection of polytopes that are all translates of the same polytope T. I will consider the following two problems.
The first is the set cover problem where we want to select a minimal number of polytopes from the given collection 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, 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.

I will give the first constant-factor approximation algorithms for both problems. The result is achieved by providing an epsilon-net for translates of a polytope in R^3 of size O(1/epsilon) which is tight up to a multiplicative constant.