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.
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Pandit, S. (2019). Covering and Packing of Triangles Intersecting a Straight Line. In: Pal, S., Vijayakumar, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2019. Lecture Notes in Computer Science(), vol 11394. Springer, Cham. https://doi.org/10.1007/978-3-030-11509-8_18
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