Covering and Packing of Triangles Intersecting a Straight Line

  • Supantha PanditEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)


We study the four geometric optimization problems: Open image in new window , Open image in new window , Open image in new window , and Open image in new window with Open image in new window (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 Open image in new window . The straight line can either be a Open image in new window or an Open image in new window line (a line whose slope is \(-1\)). A right-triangle is said to be a Open image in new window , 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.


Set cover Hitting set Piercing set Independent set Horizontal line Inclined line Diagonal line \(\mathsf {NP}\)-hard Right triangles Dynamic programming. 


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Authors and Affiliations

  1. 1.Stony Brook UniversityStony BrookUSA

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