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 $k\in \mathbb{N}$.

翻译：我们研究了多联骨牌在$k$-跳可见性模型下的艺术画廊问题。在该可见性模型中，多联骨牌的两个单位方块能够相互看见当且仅当它们在多联骨牌对偶图中对应顶点间的最短路径长度不超过$k$。本文证明该问题在简单多联骨牌中的VC维数为$3$，而在带孔多联骨牌中为$4$。此外，我们通过从平面单调3SAT问题归约，证明即使在薄多联骨牌（即不包含$2\times 2$方块块的多联骨牌）中该问题也是NP完全的。作为补充，针对所有$k\in \mathbb{N}$，我们为简单$2$-薄多联骨牌（不包含$3\times 3$方块块）提出了一个线性时间的$4$-近似算法。