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# Computing Smallest Convex Intersecting Polygons

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LIPIcs.ESA.2022.9.pdf
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## Cite As

Antonios Antoniadis, Mark de Berg, Sándor Kisfaludi-Bak, and Antonis Skarlatos. Computing Smallest Convex Intersecting Polygons. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 9:1-9:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.9

## Abstract

A polygon C is an intersecting polygon for a set O of objects in ℝ² if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• convex hull
• imprecise points
• computational geometry

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## References

1. Antonios Antoniadis, Krzysztof Fleszar, Ruben Hoeksma, and Kevin Schewior. A PTAS for Euclidean TSP with hyperplane neighborhoods. ACM Trans. Algorithms, 16(3):38:1-38:16, 2020.
2. Moshe Dror, Alon Efrat, Anna Lubiw, and Joseph SB Mitchell. Touring a sequence of polygons. In STOC 2003: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 473-482, 2003.
3. Adrian Dumitrescu. The traveling salesman problem for lines and rays in the plane. Discrete Mathematics, Algorithms and Applications, 4(04):1250044, 2012.
4. Adrian Dumitrescu and Minghui Jiang. Minimum-perimeter intersecting polygons. Algorithmica, 63(3):602-615, 2012. URL: https://doi.org/10.1007/s00453-011-9516-3.
5. Ray A. Jarvis. On the identification of the convex hull of a finite set of points in the plane. Information processing letters, 2(1):18-21, 1973.
6. Ahmad Javad, Ali Mohades, Mansoor Davoodi, and Farnaz Sheikhi. Convex hull of imprecise points modeled by segments in the plane, 2010.
7. Yiyang Jia and Bo Jiang. The minimum perimeter convex hull of a given set of disjoint segments. In International Conference on Mechatronics and Intelligent Robotics, pages 308-318. Springer, 2017.
8. Shunhua Jiang, Zhao Song, Omri Weinstein, and Hengjie Zhang. A faster algorithm for solving general lps. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing 2021, pages 823-832. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451058.
9. Marc J. van Kreveld and Maarten Löffler. Approximating largest convex hulls for imprecise points. J. Discrete Algorithms, 6(4):583-594, 2008. URL: https://doi.org/10.1016/j.jda.2008.04.002.
10. Maarten Löffler and Marc J. van Kreveld. Largest and smallest convex hulls for imprecise points. Algorithmica, 56(2):235-269, 2010. URL: https://doi.org/10.1007/s00453-008-9174-2.
11. Franco P. Preparata and Se June Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20(2):87-93, 1977.
12. Xuehou Tan. The touring rays and related problems. Theoretical Computer Science, 2021.
13. Csaba D. Tóth, Joseph O'Rourke, and Jacob E Goodman. Handbook of discrete and computational geometry. CRC press, 2017.
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