Hardness results and approximation schemes for discrete packing and domination problems

Abstract

The Maximum Independent Set(MIS) and Minimum Dominating Set (MDS) problems are well-known problems in computer science. In this paper, we consider discrete versions of both of these problems - Maximum Discrete Independent Set (MDIS) and Minimum Discrete Independent Set (MDDS). For both problems, the input is a set of geometric objects (Formula presented) and a set of points P in the plane. In the MDIS problem, the objective is to find a maximum size subset (Formula presented) of objects such that no two objects in (Formula presented) have a point in common from P. On the other hand, in the MDDS problem, the objective is to find a minimum size subset (Formula presented) such that for every object (Formula presented) there exists at least one object (Formula presented) such that O ⋂ O′ contains a point from P. In this paper, we present PTASes based on local search technique for both MDIS and MDDS problems, where the objects are arbitrary radii disks and arbitrary side length axis-parallel squares. Further, we show that the MDDS problem is APX-hard for axis-parallel rectangles, ellipses, axis-parallel strips, downward shadows of line segments, etc. in ℝ 2 and for cubes and spheres in ℝ 3 . Finally, we prove that both MDIS and MDDS problems are NP-hard for unit disks intersecting a horizontal line and for axis-parallel unit squares intersecting a straight line with slope -1.