Abstract:
In this article, we study the gradation of the complexity of the weighted set cover problem with axis-parallel rectangles whose side lengths are bounded integers. We show that the mod-one method of Chan and Hu (2015) for unit squares can be extended to these objects to get a polynomial-time approximation scheme (PTAS). We further show that the problem has a polynomial-time algorithm when all rectangles intersect a given horizontal line. On the contrary, we show that even the unweighted version of the problem is NP-hard when every rectangle intersects at least one of two given horizontal lines at the unit vertical distance.
