An algorithm for constructing regions with rectangles: Independence and minimum generating sets for collections of intervals

We provide an algorithm which solves the following problem: given a polygon with edges parallel to the x and y axes, which is convex in the y direction, find a minimum size collection of rectangles, which cover the polygon and are contained within it. The algorithm is quadratic in the number of vertices of the polygon. Our method also yields a new proof of a recent duality theorem equating minimum size rectangle covers to maximum size sets of independent points in the polygon.