Dominating set of rectangles intersecting a straight line

Abstract

The Minimum Dominating Set (MDS) problem is one of the well-studied problems in computer science. It is well-known that this problem is $$\mathsf {NP}$$ -hard for simple geometric objects; unit disks, axis-parallel unit squares, and axis-parallel rectangles to name a few. An interesting variation of the MDS problem with rectangles is when there exists a straight line that intersects each of the given rectangles. In the recent past researchers have studied the maximum independent set, minimum hitting set problems on this setting with different geometric objects. We study the MDS problem with axis-parallel rectangles, unit-height rectangles, and unit squares in the plane. These geometric objects are constrained to be intersected by a straight line. For axis-parallel rectangles, we prove that this problem is $$\mathsf {NP}$$ -hard. When the objects are axis-parallel unit squares, we present a polynomial time algorithm using dynamic programming. We provide a polynomial time algorithm for unit-height rectangles as well. For unit squares that touch the straight line at a single point from either side of the straight line, we show that there is an $$O(n\log n)$$ -time algorithm.