Covering and Packing of Triangles Intersecting a Straight Line

Abstract

We study the four geometric optimization problems: , , , and with (a triangle is a right-triangle whose base is parallel to the x-axis, perpendicular is parallel to the y-axis, and the slope of the hypotenuse is $-1$). The input triangles are constrained to be intersecting a . The straight line can either be a or an line (a line whose slope is $-1$). A right-triangle is said to be a , if the length of both its base and perpendicular is $\lambda$. For $1$-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are $\mathsf {NP}$-hard, whereas the piercing set and independent set problems are in $\mathsf {P}$. The same results hold for $1$-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are $\mathsf {NP}$-hard. Finally, we give an $n^{O(\lceil \log c\rceil +1)}$ time exact algorithm for the independent set problem with $\lambda$-right-triangles intersecting a straight line such that $\lambda$ takes more than one value from [1, c], for some integer c. We also present $O(n^2)$ time dynamic programming algorithms for the independent set problem with $1$-right-triangles where the triangles intersect a horizontal line and an inclined line.

S. Pandit—Partially supported by the Indo-US Science & Technology Forum (IUSSTF) under the SERB Indo-US Postdoctoral Fellowship scheme with grant number 2017/94, Department of Science and Technology, Government of India.