Covering and packing of triangles intersecting a straight line

Abstract

We study the four geometric optimization problems: set cover, hitting set, piercing set, and independent set with right-triangles (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 straight line. The straight line can either be a horizontal or an inclined line (a line whose slope is - 1). A right-triangle is said to be a λ-right-triangle, if the length of both its base and perpendicular is λ. For 1-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set prob- lems are NP -hard, whereas the piercing set and independent set problems are in 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 intersect- ing an inclined line are NP -hard. Finally, we give an n (Formula presented) time exact algorithm for the independent set problem with λ-right-triangles intersecting a straight line such that λ takes more than one value from [1, c], for some integer c. We also present O(n2) time dynamic programming algorithms for the independent set problem with 1-right-triangles where the triangles intersect a horizontal line and an inclined line.