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arXiv · 2308.00334

Guarding Polyominoes Under $k$-Hop Visibility

Omrit Filtser·Erik Krohn·Bengt J. Nilsson·Christian Rieck·Christiane Schmidt
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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 $k\in \mathbb{N}$.

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Omrit Filtser, Erik Krohn, Bengt J. Nilsson, Christian Rieck, Christiane Schmidt. 2023-08-01. Guarding Polyominoes Under $k$-Hop Visibility. https://arxiv.org/abs/2308.00334

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