Guarding Polyominoes under $k$-Hop Visibility
Abstract
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 $k\in \mathbb{N}$.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2023
- DOI:
- 10.48550/arXiv.2308.00334
- arXiv:
- arXiv:2308.00334
- Bibcode:
- 2023arXiv230800334F
- Keywords:
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- Computer Science - Computational Geometry;
- Computer Science - Data Structures and Algorithms;
- F.2.2
- E-Print:
- 17 pages, 11 figures. Full version of an extended abstract that has been accepted to LATIN 2024. Some parts have been improved based on reviewer comments