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Hardness Results and Approximation Schemes for Discrete Packing and Domination Problems

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

Abstract

The Open image in new window and Open image in new window problems are well-known problems in computer science. In this paper, we consider Open image in new window versions of both of these problems - Open image in new window and Open image in new window . 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 Open image in new window 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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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology RoparRupnagarIndia
  2. 2.Stony Brook UniversityStony BrookUSA

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