Approximation of Rectangle Stabbing and Interval Stabbing Problems

摘要

The weighted rectangle multi-stabbing problem (WRMS) can be described as follows: given is a grid in IR~2 consisting of columns and rows each having a positive integral weight, and a set of closed axis-parallel rectangles each having a positive integral demand. The rectangles are placed arbitrarily in the grid with the only assumption that each rectangle is intersected by at least one column and at least one row. The objective is to find a minimum weight (multi)set of columns and rows of the grid so that for each rectangle the total multiplicity of selected columns and rows stabbing it is at least its demand. (A column or row is said to stab a rectangle if it intersects it.)