Journal of Combinatorial Optimization

, Volume 3, Issue 4, pp 437–452 | Cite as

Close Approximations of Minimum Rectangular Coverings

  • Joachim Gudmundsson
  • Christos Levcopoulos


We consider the problem of covering arbitrary polygons with rectangles. The rectangles must lie entirely within the polygon. (This requires that the interior angles of the polygon are all greater than or equal to 90 degrees.) We want to cover the polygon with as few rectangles as possible. This problem has an application in fabricating masks for integrated circuits.

In this paper we will describe the first polynomial algorithm, guaranteeing an O(log n) approximation factor, provided that the n vertices of the input polygon are given as polynomially bounded integer coordinates. By the same technique we also obtain the first algorithm producing a covering which is within a constant factor of the optimal in exponential time (compared to the doubly-exponential known before).

approximation algorithms computational geometry covering polygons 


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  1. P. Berman and B. DasGupta, "On the complexities of efficient solutions of the rectilinear polygon cover problems," Algoritmica, vol. 17, pp. 331–356, 1997.Google Scholar
  2. T.H. Carmen, C.E. Leiserson, and R.L. Rivest, Introduction to Algorithms, MIT Press, 1990.Google Scholar
  3. B.M. Chazelle, "Computational geometry and convexity," PhD Thesis, Department of Computer Science, Yale University, New Haven, CT, 1979. Carnegie-Mellon Univ. Report CS-80–150.Google Scholar
  4. J.C. Culberson and R.A. Reckhow, "Covering polygon is hard," Journal of Algorithms, vol. 17, pp. 2–44, 1994.Google Scholar
  5. D. Franzblau and D. Kleitman, "An Algorithm for Constructing Regions with Rectangles," in Proceedings of the 16th Annual ACM Symposium on Theory of Computation, 1984, pp. 167–174.Google Scholar
  6. A. Heged¨us, "Algorithms for covering polygons by rectangles," Computer Aided Design, vol. 14, no. 5, 1982.Google Scholar
  7. J.M. Keil, "Minimally Covering a Horizontally Convex Orthogonal Polygon," in Proceedings of the 2nd Annual ACM Symposium on Computational Geometry, 1986, pp. 43–51.Google Scholar
  8. C. Levcopoulos, "A Fast Heuristic for Covering Polygons by Rectangles," in Proceedings of Fundamentals of Computation Theory, LNCS, vol. 199, 1985.Google Scholar
  9. C. Levcopoulos, "Improved Bounds for Covering General Polygons with Rectangles, in Proceedings of the 7th Foundations of Software Technology and Theoretical Computer Science, LNCS, vol. 287, 1987.Google Scholar
  10. L. Monk, "Elementary-recursive decision procedures," PhD Thesis, University of California, Berkley, 1975.Google Scholar
  11. J. O'Rourke, "The complexity of computing minimum convex covers for polygons," Report JHU-EE 82–1, Department of Electrical Engineering and Computer Science, The Johns Hopkins University, Baltimore, MD, 1982.Google Scholar
  12. J. O'Rourke and K.J. Supowit, "Some NP-hard polygon decomposition problems," IEEE Transactions on Information Theory, vol. IT-29, pp. 181–190, 1983.Google Scholar
  13. M. Yamashita, T. Ibaraki, and N. Honda, "The minimum number cover problem of a rectilinear region by rectangles," EATCS, vol. 24, p. 138, 1984.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Joachim Gudmundsson
    • 1
  • Christos Levcopoulos
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden

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