On the geometric red-blue set cover problem

Article Type

Research Article

Publication Title

Theoretical Computer Science

Abstract

Using various geometric objects, we study variations of the geometric Red-Blue Set Cover (RBSC) problem in R2. Here, given two sets R and B of points, called red and blue points, respectively, and a set O of objects, the objective is to compute a subset O′⊆O of objects such that O′ covers all the blue points in B and covers the minimum number of red points in R. We show that the RBSC problem with intervals on the real line is polynomial-time solvable. In R2, the problem admits a polynomial-time algorithm for a particular case of axis-parallel lines when no two lines intersect at a red point. However, if the objects are horizontal lines and vertical segments, the problem becomes NP-hard. It remains NP-hard for axis-parallel unit length segments distributed arbitrarily in the plane. The problem is NP-hard for the axis-parallel rectangles even when (i) each rectangle in O is anchored at one of the given two parallel lines, and (ii) a horizontal line intersects all the rectangles in O. We show a variation of RBSC, called Special Red Blue Set Cover (SPECIAL-RBSC), to be APX-hard. Next, we use this result to show APX-hardness of the following geometric variations of RBSC problem in R2 where the objects are (i) axis-parallel rectangles containing the origin, (ii) axis-parallel strips, (iii) axis-parallel rectangles that are intersecting exactly zero or four times, (iv) axis-parallel line segments, and (v) downward shadows of line segments, by providing the encoding of these problems as the Special Red Blue Set Cover problem. These APX-hardness results are in the same line of work by Chan and Grant (2014), who provided the APX-hardness results for the geometric set cover problem for the above classes of objects.

DOI

10.1016/j.tcs.2025.115089

Publication Date

4-7-2025

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