Hardness Results and Approximation Schemes for Discrete Packing and Domination Problems

Abstract

The and problems are well-known problems in computer science. In this paper, we consider versions of both of these problems - and . For both problems, the input is a set of geometric objects $\mathcal {O}$ and a set of points $\mathcal {P}$ in the plane. In the MDIS problem, the objective is to find a maximum size subset $\mathcal {O}^\prime \subseteq \mathcal {O}$ of objects such that no two objects in $\mathcal {O}^\prime$ have a point in common from $\mathcal {P}$. On the other hand, in the MDDS problem, the objective is to find a minimum size subset $\mathcal {O}^\prime \subseteq \mathcal {O}$ such that for every object $O \in \mathcal {O} \setminus \mathcal {O}^\prime$ there exists at least one object $O^\prime \in \mathcal {O}^\prime$ such that $O\,\cap \,O^\prime$ contains a point from $\mathcal {P}$.

In this paper, we present $\mathsf {PTAS}$es based on 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 $\mathsf {APX}$-hard for axis-parallel rectangles, ellipses, axis-parallel strips, downward shadows of line segments, etc. in $\mathbb {R}^2$ and for cubes and spheres in $\mathbb {R}^3$. Finally, we prove that both MDIS and MDDS problems are $\mathsf {NP}$-hard for unit disks intersecting a horizontal line and for axis-parallel unit squares intersecting a straight line with slope $-1$.

S. Pandit—The author is partially supported by the Indo-US Science & Technology Forum (IUSSTF) under the SERB Indo-US Postdoctoral Fellowship scheme with grant number 2017/94, Department of Science and Technology, Government of India.