Covering and Packing of Triangles Intersecting a Straight Line

We study the four geometric optimization problems: set cover, hitting set, piercing set, andindependent 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 linewhose slope is - 1). A right-triangle is said to be a lambda-right-triangle, 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 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 intersecting an inclined line are NP-hard. Finally, we give an n(O((sic)log c(sic)+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.