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Complexities of efficient solutions of rectilinear polygon cover problems

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Abstract

The rectilinear polygon cover problem is one in which a certain class of features of a rectilinear polygon ofn vertices has to be covered with the minimum number of rectangles included in the polygon. In particular, we consider covering the entire interior, the boundary, and the set of corners of the polygon. These problems have important applications in storing images and in the manufacture of integrated circuits. Unfortunately, most of these problems are known to be NP-complete. Hence it is necessary to develop efficient heuristics for these problems or to show that the design of efficient heuristics is impossible. In this paper we show:

  1. (a)

    The corner cover problem is NP-complete.

  2. (b)

    The boundary and the corner cover problem can be approximated within a ratio of 4 of the optimum inO(n logn) andO(n 1.5) time, respectively.

  3. (c)

    No polynomial-time approximation scheme exists for the interior and the boundary cover problems, unless P=NP.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Pennsylvania State University, 16802, University Park, PA, USA

    P. Berman

  2. Department of Computer Science, University of Minnesota, 55455-0159, Minneapolis, MN, USA

    B. DasGupta

Authors
  1. P. Berman
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  2. B. DasGupta
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Additional information

Communicated by C. K. Yap.

A preliminary version of this result appeared inProceedings of the Fourth Canadian Conference on Computational Geometry, 1992, pp. 229–235. This research was partially supported by NSF Grant CCR-9114545.

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Berman, P., DasGupta, B. Complexities of efficient solutions of rectilinear polygon cover problems. Algorithmica 17, 331–356 (1997). https://doi.org/10.1007/BF02523677

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  • DOI: https://doi.org/10.1007/BF02523677

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