Enclosing Points with Geometric Objects

Document detail

ID

oai:arXiv.org:2402.17322

Topic

Computer Science - Computational G... Computer Science - Data Structures...

Author

Chan, Timothy M. He, Qizheng Xue, Jie

Year

2024

listing date

3/6/2024

Keywords

$o $-approximation

Metrics

Abstract

Let $X$ be a set of points in $\mathbb{R}^2$ and $\mathcal{O}$ be a set of geometric objects in $\mathbb{R}^2$, where $|X| + |\mathcal{O}| = n$.

We study the problem of computing a minimum subset $\mathcal{O}^* \subseteq \mathcal{O}$ that encloses all points in $X$.

Here a point $x \in X$ is enclosed by $\mathcal{O}^*$ if it lies in a bounded connected component of $\mathbb{R}^2 \backslash (\bigcup_{O \in \mathcal{O}^*} O)$.

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 $O(\alpha(n)\log n)$-approximation algorithm for segments, where $\alpha(n)$ is the inverse Ackermann function, and an $O(\log n)$-approximation algorithm for disks.

;Comment: In SoCG'24

Chan, Timothy M.,He, Qizheng,Xue, Jie, 2024, Enclosing Points with Geometric Objects