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Decomposing a Simple Polygon with Geodesic Unit-Balls

Authors Reilly Browne , Prahlad Narasimhan Kasthurirangan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Packing and covering problems
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Covering
  • partitioning
  • polygon
  • k-center
  • constant factor approximation

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Abstract

We consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the 𝓁₂-metric for distances. There is no known method for finding an exact solution to these problems, even when the input size is constant, and the problem is known to be NP-hard in the case of polygons with holes. With this in mind, we instead devote our attention to developing simple approximation algorithms that run in polynomial time. For the radius problem, we present the first known approximation algorithms for both covering and partitioning, achieving a factor of 9. For the diameter problem, we are only able to give a positive result for the partition version of the problem, where we improve upon a complicated 72-approximation from Abrahamsen and Rasmussen [Mikkel Abrahamsen and Nichlas Langhoff Rasmussen, 2025], achieving a simple 15-approximation.

Cite As Get BibTex

Reilly Browne and Prahlad Narasimhan Kasthurirangan. Decomposing a Simple Polygon with Geodesic Unit-Balls. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.67

Author Details

Reilly Browne
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Prahlad Narasimhan Kasthurirangan
  • Department of Applied Mathematics and Statistics, Stony Brook University, NY, USA

Funding

  • Kasthurirangan, Prahlad Narasimhan: Supported by the US Office of Naval Research under grant no. N00014-26-1-2088.

Acknowledgements

We would like to thank the anonymous reviewers of ESA 2026 for their valuable feedback.

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