# Hardness Results and Approximation Schemes for Discrete Packing and Domination Problems

• Apurva Mudgal
• Supantha Pandit
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

## 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 $$-1$$.

## Keywords

Discrete Independent Set Discrete Dominating Set Local search $$\mathsf {PTAS}$$ $$\mathsf {NP}$$-hard $$\mathsf {APX}$$-hard Disks Axis-parallel squares Axis-parallel rectangles

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