eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
35:1
35:15
10.4230/LIPIcs.SoCG.2024.35
article
Enclosing Points with Geometric Objects
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
He, Qizheng
1
https://orcid.org/0000-0002-2518-1114
Xue, Jie
2
https://orcid.org/0000-0001-7015-1988
Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Department of Computer Science, New York University Shanghai, China
Let X be a set of points in βΒ² and πͺ be a set of geometric objects in βΒ², where |X| + |πͺ| = n. We study the problem of computing a minimum subset πͺ^* β πͺ that encloses all points in X. Here a point x β X is enclosed by πͺ^* if it lies in a bounded connected component of βΒ²β(β_{O β πͺ^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(Ξ±(n)log n)-approximation algorithm for segments, where Ξ±(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.35/LIPIcs.SoCG.2024.35.pdf
obstacle placement
geometric optimization
approximation algorithms