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Smallest Convex Hulls of Polygons

Authors Mook Kwon Jung, Hee-Kap Ahn

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Convex hull
  • packing
  • bundling
  • approximation algorithms
  • cuttings

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Abstract

We study the problem of minimizing the area of the convex hull of k polygons with a total of n vertices in the plane, under translations and rigid motions for any fixed k ≥ 3. For any ε ∈ (0, 1), we give (1 + ε)-approximation algorithms running in O(ε^{-1/2} log n + ε^{1/2 - k}) time for translations, and in O(ε^{-1/2} log n + ε^{3/2 - 2k}) time for rigid motions. We also consider minimizing the perimeter of the convex hull under translations and obtain a (1 + ε)-approximation algorithm running in O(ε^{-1/2} log n + ε^{1/2 - k}log^{k-1}(1/ε)) time. To the best of our knowledge, these are the first results of this kind for k ≥ 3 polygons. Furthermore, for the special case of two polygons with n₀ and n₁ vertices (n₀ ≥ n₁), respectively, we give an O(n₀+n₁log²(n₀+n₁))-time algorithm for the minimum-perimeter problem. This significantly improves upon the best-known O((n₀+n₁)log²(n₀+n₁)) bound by eliminating the logarithmic overhead associated with the larger input size n₀.

Cite As Get BibTex

Mook Kwon Jung and Hee-Kap Ahn. Smallest Convex Hulls of Polygons. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.44

Author Details

Mook Kwon Jung
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Hee-Kap Ahn
  • Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea

Funding

This research was supported by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (Nos. 2017-0-00905 and RS-2024-00439856, Software Star Lab), and (No. RS-2019-II191906, Artificial Intelligence Graduate School Program(POSTECH)). This research was also partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2023-00219980).

Acknowledgements

We would like to thank the anonymous reviewer for their careful and constructive review.

References

  1. Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. Journal of the ACM, 51(4):606-635, 2004. URL: https://doi.org/10.1145/1008731.1008736.
  2. Hee-Kap Ahn, Judit Abardia, Sang Won Bae, Otfried Cheong, Susanna Dann, Dongwoo Park, and Chan-Su Shin. The minimum convex container of two convex polytopes under translations. Computational Geometry, 77:40-50, 2019. URL: https://doi.org/10.1016/j.comgeo.2018.02.004.
  3. Hee-Kap Ahn, Peter Brass, Otfried Cheong, Hyeon-Suk Na, Chan-Su Shin, and Antoine Vigneron. Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets. Computational Geometry, 33(3):152-164, 2006. URL: https://doi.org/10.1016/j.comgeo.2005.06.001.
  4. Hee-Kap Ahn, Peter Brass, and Chan-Su Shin. Maximum overlap and minimum convex hull of two convex polyhedra under translations. Computational Geometry, 40(2):171-177, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.08.001.
  5. Hee-Kap Ahn, Siu-Wing Cheng, and Iris Reinbacher. Maximum overlap of convex polytopes under translation. Computational Geometry, 46(5):552-565, 2013. URL: https://doi.org/10.1016/j.comgeo.2011.11.003.
  6. Hee-Kap Ahn and Otfried Cheong. Aligning two convex figures to minimize area or perimeter. Algorithmica, 62(1):464-479, 2012. URL: https://doi.org/10.1007/s00453-010-9466-1.
  7. Helmut Alt, Mark de Berg, and Christian Knauer. Approximating minimum-area rectangular and convex containers for packing convex polygons. Journal of Computational Geometry, 8(1):1-10, 2017. URL: https://doi.org/10.20382/jocg.v8i1a1.
  8. Helmut Alt and Ferran Hurtado. Packing convex polygons into rectangular boxes. In Proceedings of Japanese Conference on Discrete and Computational Geometry, pages 67-80. Springer, 2000. URL: https://doi.org/10.1007/3-540-47738-1_5.
  9. Timothy M Chan and Isaac M Hair. A linear time algorithm for the maximum overlap of two convex polygons under translation. In Proceedings of the 41st International Symposium on Computational Geometry, pages 31-1, 2025. URL: https://doi.org/10.4230/LIPIcs.SoCG.2025.31.
  10. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete & Computational Geometry, 9(2):145-158, 1993. URL: https://doi.org/10.1007/BF02189314.
  11. Jongmin Choi, Dongwoo Park, and Hee-Kap Ahn. Bundling two simple polygons to minimize their convex hull. In Proceedings of the 11th International Workshop on Algorithms and Computation, pages 66-77. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-53925-6_6.
  12. Mark de Berg, Otfried Cheong, Olivier Devillers, Marc van Kreveld, and Monique Teillaud. Computing the maximum overlap of two convex polygons under translations. Theory of Computing Systems, 31(5):613-628, 1998. URL: https://doi.org/10.1007/PL00005845.
  13. Mook Kwon Jung, Seokyun Kang, and Hee-Kap Ahn. Minimum convex hull and maximum overlap of two convex polytopes. In Proceedings of the 36th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 3547-3561. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.117.
  14. Vladlen Koltun. Almost tight upper bounds for vertical decompositions in four dimensions. Journal of the ACM, 51(5):699-730, 2004. URL: https://doi.org/10.1145/1017460.1017461.
  15. Hyun-Chan Lee and Tony C Woo. Determining in linear time the minimum area convex hull of two polygons. IIE transactions, 20(4):338-345, 1988. Google Scholar
  16. David M Mount, Ruth Silverman, and Angela Y Wu. On the area of overlap of translated polygons. Computer Vision and Image Understanding, 64(1):53-61, 1996. URL: https://doi.org/10.1006/cviu.1996.0045.
  17. Dongwoo Park, Sang Won Bae, Helmut Alt, and Hee-Kap Ahn. Bundling three convex polygons to minimize area or perimeter. Computational Geometry, 51:1-14, 2016. URL: https://doi.org/10.1016/j.comgeo.2015.10.003.
  18. Kai Tang, Charlie CL Wang, and Danny Z Chen. Minimum area convex packing of two convex polygons. International Journal of Computational Geometry & Applications, 16(01):41-74, 2006. URL: https://doi.org/10.1142/S0218195906001926.
  19. Isaak M. Yaglom and Vladimir G. Boltyansky. Convex Figures. Holt, Rinehart and Winston, New York, 1961. Google Scholar
  20. Günter M. Ziegler. Lectures on Polytopes. Springer New York, 1995. Google Scholar
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