A Deterministic Bicriteria Approximation Algorithm for the Art Gallery Problem
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917409973075968 |
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| author | Elbassioni, Khaled |
| author_facet | Elbassioni, Khaled |
| contents | Given a polygon $H$ in the plane, the art gallery problem calls for fining the smallest set of points in $H$ from which every other point in $H$ is seen. We give a deterministic algorithm that, given any polygon $H$ with $h$ holes, $n$ rational veritces of maximum bit-length $L$, and a parameter $δ\in(0,1)$, is guaranteed to find a set of points in $H$ of size $O\big(\OPT\cdot\log(h+2)\cdot\log (\OPT\cdot\log(h+2)))$ that sees at least a $(1-δ)$-fraction of the area of the polygon. The running time of the algorithm is polynomial in $h$, $n$, $L$ and $\log(\frac{1}δ)$, where $\OPT$ is the size of an optimum solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23297 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Deterministic Bicriteria Approximation Algorithm for the Art Gallery Problem Elbassioni, Khaled Computational Geometry Discrete Mathematics Data Structures and Algorithms F.2.2 Given a polygon $H$ in the plane, the art gallery problem calls for fining the smallest set of points in $H$ from which every other point in $H$ is seen. We give a deterministic algorithm that, given any polygon $H$ with $h$ holes, $n$ rational veritces of maximum bit-length $L$, and a parameter $δ\in(0,1)$, is guaranteed to find a set of points in $H$ of size $O\big(\OPT\cdot\log(h+2)\cdot\log (\OPT\cdot\log(h+2)))$ that sees at least a $(1-δ)$-fraction of the area of the polygon. The running time of the algorithm is polynomial in $h$, $n$, $L$ and $\log(\frac{1}δ)$, where $\OPT$ is the size of an optimum solution. |
| title | A Deterministic Bicriteria Approximation Algorithm for the Art Gallery Problem |
| topic | Computational Geometry Discrete Mathematics Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2512.23297 |