Improved Results on Geometric Hitting Set Problems

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ℝ3 and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.