Guarding Polyominoes Under $k$-Hop Visibility

TL;DR

This paper investigates the $k$-hop visibility Art Gallery Problem in polyominoes, establishing VC dimension bounds, NP-completeness, and providing a linear-time approximation algorithm for specific polyomino classes.

Contribution

It introduces VC dimension bounds for $k$-hop visibility in polyominoes, proves NP-completeness in thin polyominoes, and offers a linear-time approximation algorithm for simple $2$-thin polyominoes.

Findings

VC dimension is 3 in simple polyominoes

NP-complete in thin polyominoes

Linear-time 4-approximation algorithm for simple 2-thin polyominoes

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 $k\in \mathbb{N}$.