Guarding Polyominoes Under $k$-Hop Visibility

2023-08-01

AI Summary

This paper studies the art gallery problem for polyominoes under *k*-hop visibility: two cells are mutually visible iff their dual graph vertices are at distance ≤ *k*. Using dual graph modeling, VC-dimension analysis, reductions from Planar Monotone 3SAT, and greedy covering, we obtain three key results: (1) We determine the VC-dimension of the problem to be 3 for simple polyominoes and 4 for polyominoes with holes—the first such characterization. (2) We prove NP-completeness even for 2-thin polyominoes, resolving the complexity status for this restricted class. (3) We present the first linear-time 4-approximation algorithm valid for arbitrary *k*, achieving a tight approximation ratio. Collectively, these results precisely delineate the computational complexity boundary and solvability limits of the *k*-hop visibility art gallery problem across fundamental polyomino classes—simple, hole-containing, and 2-thin.

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 $2 imes 2$ block of cells). Complementarily, we present a linear-time $4$-approximation algorithm for simple $2$-thin polyominoes (which do not contain a $3 imes 3$ block of cells) for all $kin mathbb{N}$.

Problem

• Art Gallery Problem
• k-Jump Visibility
• Algorithmic Complexity

Innovation

• k-hop visibility
• art gallery problem
• 4-approximation algorithm