Rectangle Covers Revisited Computationally

Abstract

We consider the problem of covering an orthogonal polygon with a minimum number of axis-parallel rectangles from a computational point of view. We propose an integer program which is the first general approach to obtain provably optimal solutions to this well-studied ${\mathcal NP}$-hard problem. It applies to common variants like covering only the corners or the boundary of the polygon, and also to the weighted case. In experiments it turns out that the linear programming relaxation is extremely tight, and rounding a fractional solution is an immediate high quality heuristic. We obtain excellent experimental results for polygons originating from VLSI design, fax data sheets, black and white images, and for random instances. Making use of the dual linear program, we propose a stronger lower bound on the optimum, namely the cardinality of a fractional stable set. We outline ideas how to make use of this bound in primal-dual based algorithms. We give partial results which make us believe that our proposals have a strong potential to settle the main open problem in the area: To find a constant factor approximation algorithm for the rectangle cover problem.