# Maximum Independent and Disjoint Coverage

• Amit Kumar Dhar
• Supantha Pandit
• Jagpreet Singh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

## Abstract

Set Cover is one of the most studied optimization problems in Computer Science. In this paper, we target two interesting variations of this problem in a geometric setting: (i) , and (ii) problems. In both problems, the input consists of a set P of points and a set O of geometric objects in the plane. The objective is to maximize the number of points covered by a set $$O'$$ of selected objects from O. In the MDC problem we restrict the objects in $$O'$$ are pairwise disjoint (non-intersecting). Whereas, in the MIC problem any pair of objects in $$O'$$ should not share a point from P (however, they may intersect each other). We consider various geometric objects as covering objects such as axis-parallel infinite lines, axis-parallel line segments, unit disks, axis-parallel unit squares, and intervals on a real line. For axis-parallel infinite lines both MDC and MIC problems admit polynomial time algorithms. On the other hand, we prove that the MIC problem is $$\mathsf {NP}$$-complete when the objects are horizontal infinite lines and vertical segments. We also prove that both MDC and MIC problems are $$\mathsf {NP}$$-complete for axis-parallel unit segments in the plane. For unit disks and axis-parallel unit squares, we prove that both these problems are $$\mathsf {NP}$$-complete. Further, we present $$\mathsf {PTAS}$$es for the MDC problem for unit disks as well as unit squares using Hochbaum and Maass’s “shifting strategy”. For unit squares, we design a $$\mathsf {PTAS}$$ for the MIC problem using Chan and Hu’s “mod-one transformation” technique. In addition to that, we give polynomial time algorithms for both MDC and MIC problems with intervals on the real line.

## Keywords

Set cover Maximum coverage Independent set $$\mathsf {NP}$$-hard $$\mathsf {PTAS}$$ Line Segment Disk Square

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## Authors and Affiliations

• Amit Kumar Dhar
• 1
• 2
• Supantha Pandit
• 3
Email author
• Jagpreet Singh
• 4
1. 1.Department of Electrical Engineering and Computer ScienceIndian Institute of Technology BhilaiDatrengaIndia
2. 2.Department of Computer Science and EngineeringIndian Institute of Technology RoparRupnagarIndia
3. 3.Stony Brook UniversityStony BrookUSA