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- W4313350034 abstract "We study geometric variations of the Red-Blue Hitting Set problem. Given two sets of objects R and B, colored red and blue, respectively, and a set of points P in the plane, the goal is to find a subset $$P^prime subseteq P$$ of points that hits all blue objects in B while hitting the minimum number of red objects in R. We study this problem for various geometric objects. We present a polynomial-time algorithm for the problem with intervals on the real line. On the other hand, we show that the problem is $$textsf{NP}$$ -hard for axis-parallel unit segments. Next, we study the problem with axis-parallel rectangles. We give a polynomial-time algorithm for the problem when the rectangles are anchored on a horizontal line. The problem is shown to be $$textsf{NP}$$ -hard when all the rectangles intersect a horizontal line. Finally, we prove that the problem is $$textsf{APX}$$ -hard when the objects are axis-parallel rectangles containing the origin of the plane, axis-parallel rectangles where every two rectangles intersect exactly either zero or four times, axis-parallel line segments, axis-parallel strips, and downward shadows of segments. To achieve these $$textsf{APX}$$ -hardness results, we first introduce a variation of the Red-Blue Hitting Set problem in a set system, called the Special-Red-Blue Hitting Set problem. We prove that the Special-Red-Blue Hitting Set problem is $$textsf{APX}$$ -hard and we provide an encoding of each class of objects mentioned above as the Special-Red-Blue Hitting Set problem." @default.
- W4313350034 created "2023-01-06" @default.
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- W4313350034 date "2022-01-01" @default.
- W4313350034 modified "2023-10-16" @default.
- W4313350034 title "Exact Algorithms and Hardness Results for Geometric Red-Blue Hitting Set Problem" @default.
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- W4313350034 doi "https://doi.org/10.1007/978-3-031-20796-9_13" @default.
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