PTAS for geometric hitting set problems via local search

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 halfspaces 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. Categories and Subject Descriptors G.2 [Combinatorial Algorithms] General Terms Algorithms, Theory Keywords Hitting Sets, Epsilon Nets, Local Search, Approximation Algorithm

• Type: Conference
• Year: 2009
• Where: COMPGEOM
• Authors: Nabil H. Mustafa, Saurabh Ray
• Keywords: COMPGEOM 2009, Discrete Geometry, Geometric Objects, Hitting Sets, Minimum Geometric Hitting