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Covering Simple Orthogonal Polygons with Rectangles

Author Aniket Basu Roy

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Polygon Covering
  • Approximation Algorithms
  • Orthogonal Polygons
  • Rectangles
  • Local Search
  • Planar Supports

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Abstract

We study the problem of Covering Orthogonal Polygons with Rectangles, focusing on three variants: covering the interior, the boundary, and the corners. While previous work provided constant-factor approximation algorithms for these problems, significant improvements had not been achieved for over two decades. The main contribution of this work is the development of a Polynomial Time Approximation Scheme (PTAS) for both the Boundary Cover and Corner Cover problems on simple polygons, using a local search algorithm. Our work advances the state of the art, improving upon the previous best-known 4-approximation for the Boundary Cover and 2-approximation for the Corner Cover problems.
The technical core of our work lies in proving the existence of planar support graphs for certain geometric hypergraphs defined by the polygon and its containment-maximal rectangles. This structural insight enables the application of the local search framework to achieve the PTAS results. We also demonstrate the limitations of this approach by constructing instances where local search fails for the Interior Cover and certain dual problems, such as the Maximum Antirectangle and Hitting Set problems. Additionally, the methods yield a PTAS for a special case of the Discrete Independent Set problem for rectangles. These results not only settle longstanding open questions but also introduce new techniques that may be of independent interest within computational geometry.

Cite As Get BibTex

Aniket Basu Roy. Covering Simple Orthogonal Polygons with Rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.2

Author Details

Aniket Basu Roy
  • Department of Computer Science and Information Sciences, Birla Institute of Technology and Science, Pilani, K K Birla Goa Campus, Zuarinagar, Sancoale, Goa, India

Funding

  • Basu Roy, Aniket: BITS Pilani NFSG ref. N5/25/1052

Acknowledgements

The author is thankful to the Theory group at Aarhus, particularly to Peyman Afshani and Chris Schwiegelshohn. Also, thanks to Sourav Chakraborty and Abhiruk Lahiri for their valuable feedback, which improved the presentation of the paper. Lastly, the author would like to thank Nguyễn Kim Thắng for insightful discussions on the Interior Cover problem during the pandemic days.

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