Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems

In the weighted rectangle stabbing problem we are given a grid in ]R2 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 special case of this problem arises when each rectangle is intersected by exactly one row. We describe two algorithms, called STAB and ROUND, that are shown to be constant-factor approximation algorithms for different variants of this stabbing problem.