Guarding Polyominoes under $k$-Hop Visibility

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 th...

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Bibliographic Details
Main Authors Filtser, Omrit, Krohn, Erik, Nilsson, Bengt J, Rieck, Christian, Schmidt, Christiane
Format Journal Article
LanguageEnglish
Published 01.08.2023
Subjects
Computer Science - Computational Geometry
Computer Science - Data Structures and Algorithms
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Summary: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 $k\in \mathbb{N}$.
DOI:10.48550/arxiv.2308.00334