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) Open image in new window , and (ii) Open image in new window 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.
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