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Enclosing Points with Geometric Objects

Authors Timothy M. Chan , Qizheng He , Jie Xue

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Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Qizheng He
  • Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

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Timothy M. Chan, Qizheng He, and Jie Xue. Enclosing Points with Geometric Objects. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.35

Abstract

Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} 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(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • obstacle placement
  • geometric optimization
  • approximation algorithms

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