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 NPcomplete even in thin polyominoes (i.e., polyominoes that do not a contain a $2\times 2$ block of cells). Complementarily, we present a lineartime $4$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:

arXiv eprints
 Pub Date:
 August 2023
 DOI:
 10.48550/arXiv.2308.00334
 arXiv:
 arXiv:2308.00334
 Bibcode:
 2023arXiv230800334F
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms;
 F.2.2
 EPrint:
 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