Rectangle Covers Revisited Computationally LAURA HEINRICH-LITAN Robert Bosch GmbH and MARCO E. L ¨ UBBECKE Technische Universit¨ at Berlin 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 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. Furthermore, 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. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Geometrical problems and computations General Terms: Algorithms, Experimentation Additional Key Words and Phrases: Linear programming, integer programming 1. INTRODUCTION A polygon with all edges either horizontal or vertical is called orthogonal. Given an orthogonal polygon P , the rectangle cover problem is to find a minimum number of possibly overlapping axis-parallel rectangles whose union is exactly An extended abstract of this paper appeared in the Proceedings of the 4th International Workshop on Efficient and Experimental Algorithms (WEA05) [Heinrich-Litan and L ¨ ubbecke 2005]. Authors’ addresses: Laura Heinrich-Litan, Robert Bosch GmbH, Automotive Equipment, Divi- sion CM Car Multimedia, Department CM-DI/EAP, Daimlerstrasse 6, D-71229 Leonberg; email: lheinrich-litan@de.adit-jv.com, Laura.Heinrich-Litan@de.bosch.com; Marco E. L¨ ubbecke, Technis- che Universit ¨ at Berlin, Institut f ¨ ur Mathematik, Sekr. MA 6-1, Straße des 17. Juni 136, D-10623 Berlin, Germany; email: m.luebbecke@math.tu-berlin.de. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or direct commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or permissions@acm.org. C  2006 ACM 1084-6654/2006/0001-ART2.6 $5.00 DOI 10.1145/1187436.1216583 http://doi.acm.org 10.1145/1187436.1216583 ACM Journal of Experimental Algorithmics, Vol. 11, Article No. 2.6, 2006, Pages 1–21.