Guarding Polyominoes Under $k$-Hop Visibility
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916540750757888 |
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| author | Filtser, Omrit Krohn, Erik Nilsson, Bengt J. Rieck, Christian Schmidt, Christiane |
| author_facet | Filtser, Omrit Krohn, Erik Nilsson, Bengt J. Rieck, Christian Schmidt, Christiane |
| contents | We study the Art Gallery Problem under $k$-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the polyomino has length at most $k$.
In this paper, we show that the VC dimension of this problem is $3$ in simple polyominoes, and $4$ in polyominoes with holes. Furthermore, we provide a reduction from Planar Monotone 3Sat, thereby showing that the problem is NP-complete even in thin polyominoes (i.e., polyominoes that do not a contain a $2\times 2$ block of cells). Complementarily, we present a linear-time $4$-approximation algorithm for simple $2$-thin polyominoes (which do not contain a $3\times 3$ block of cells) for all $k\in \mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Guarding Polyominoes Under $k$-Hop Visibility Filtser, Omrit Krohn, Erik Nilsson, Bengt J. Rieck, Christian Schmidt, Christiane Computational Geometry Data Structures and Algorithms F.2.2 We study the Art Gallery Problem under $k$-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the polyomino has length at most $k$. In this paper, we show that the VC dimension of this problem is $3$ in simple polyominoes, and $4$ in polyominoes with holes. Furthermore, we provide a reduction from Planar Monotone 3Sat, thereby showing that the problem is NP-complete even in thin polyominoes (i.e., polyominoes that do not a contain a $2\times 2$ block of cells). Complementarily, we present a linear-time $4$-approximation algorithm for simple $2$-thin polyominoes (which do not contain a $3\times 3$ block of cells) for all $k\in \mathbb{N}$. |
| title | Guarding Polyominoes Under $k$-Hop Visibility |
| topic | Computational Geometry Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2308.00334 |