The class cover problem with boxes

Abstract

In this paper we study the following problem: Given sets R and B of r red and b blue points respectively in the plane, find a minimum-cardinality set H of axis-aligned rectangles (boxes) so that every point in B is covered by at least one rectangle of H, and no rectangle of H contains a point of R. We prove the NP-hardness of the stated problem, and give either exact or approximate algorithms depending on the type of rectangles considered. If the covering boxes are vertical or horizontal strips we give an efficient algorithm that runs in O(r log r + b log b + √ rb) time. For covering with oriented halfstrips an optimal O((r + b) log(min{r, b}))-time algorithm is shown. We prove that the problem remains NP-hard if the covering boxes are half-strips oriented in any of the four orientations, and show that there exists an O(1)-approximation algorithm. We also give an NP-hardness proof if the covering boxes are squares. In this situation, we show that there exists an O(1)-approximation algorithm.