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) maximum disjoint coverage (MDC), and (ii) maximum independent coverage (MIC) 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 the covering objects axis-parallel infinite lines, we show that both MDC and MIC problems admit polynomial time algorithms. In addition to that, we give polynomial time algorithms for both MDC and MIC problems with intervals on the real line. 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.

中文翻译：

---

最大的独立和分散覆盖

集覆盖是计算机科学中研究最多的优化问题之一。在本文中，我们针对此问题在几何设置中的两个有趣的变化：（i）最大不交叠覆盖率（MDC）和（ii）最大独立覆盖率（MIC）问题。在这两个问题中，输入均包含一组点P和一组平面中的几何对象O。的目标是最大化由一组覆盖点的数量\（O'\）从所选对象的Ò。在MDC 问题中，我们限制\（O'\）中的对象成对不相交（不相交）。而在MIC 问题\（O'\）中的任何一对对象都不应共享P点（但是，它们可能彼此相交）。我们将各种几何对象视为覆盖对象，例如平行轴无限线，平行轴线段，单位圆盘，平行轴单位正方形以及实线上的间隔。对于覆盖对象的轴平行无限线，我们证明MDC 和MIC 问题均采用多项式时间算法。除此之外，我们还提供了针对MDC 和MIC 问题的多项式时间算法，其中间隔在实线上。另一方面，我们证明MIC 问题是\（{\ mathsf {NP}} \）-当对象为水平无限线和垂直线段时完成。我们还证明，对于平面中轴平行的单元段，MDC 和MIC 问题都是\（{\ mathsf {NP}} \） -完全的。对于单位圆盘和与轴平行的单位平方，我们证明这两个问题都是\（{\ mathsf {NP}} \）-完全。此外，我们 使用Hochbaum和Maass的“转移策略”提出了单位磁盘MDC问题的\（{\ mathsf {PTAS}} \） es 。对于单位平方，我们 使用Chan和Hu的“模一变换”技术 为MIC问题设计\（{\ mathsf {PTAS}} \\）。