Guarding Polyominoes Under $k$-Hop Visibility

TL;DR

The paper studies guarding polyominoes under $k$-hop visibility, formalizing the Minimum $k$-Hop Guarding Problem (M$k$GP) and its grid-graph analogue (M$k$DSP). It establishes tight VC-dimension bounds, proving $\mathrm{VC}(k)$ equals $3$ for simple polyominoes and $4$ for polyominoes with holes, and shows NP-completeness for $1$-thin polyominoes with holes even when $k\ge 2$ via Planar Monotone 3-SAT reductions. It also provides a linear-time $4$-approximation for simple $2$-thin polyominoes across all $k$ by constructing a skeleton tree and associating square blocks, together with an accompanying witness-set argument to bound optimality. Collectively, these results delineate the computational and approximation landscape of guarding under $k$-hop visibility in constrained grid-based domains, with practical implications for discrete guard placement and related domination problems in planar graphs. The techniques combine combinatorial geometry, graph-theoretic skeletons, and a reduction-based hardness framework to map the complexity frontier of this visibility model.

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 $k\in \mathbb{N}$.

Figures (3)

• Figure 1: A unit square in green with its $k$-hop-visibility region for $k=6$ (shaded in dark green) within a polyomino---a subset of the diamond shown in light green.
• Figure 6: The $k$-hop-visibility regions ($k=6$) of two guards intersect more than twice.
• Figure 13: Example for our algorithm for $k=3$: (a) Polyomino $P$, in black, with the associated vertices: vertices in $I$ in blue ($\bullet$), vertices in $B$ in red ($\bullet$), the root $r$ is indicated in magenta ($\mathbin{\vcenter{\hbox{$\bullet$}}}$), the trees $T_I$ and $T_B$ with edges in $E_I$ are shown in blue, edges in $E_B$ are shown in red; (b) connecting $T_I$ and $T_B$, vertices $x_{v,u}$ and their incident edges are shown in green ($\bullet$), all other connecting edges are shown in black; (c) unit squares added to $D$ in light pink; (d) placement of witnesses from the proof of \ref{['th:4-appx']} in turquoise.

Theorems & Definitions (6)

• Proposition 3
• Lemma 5: Rest-Budget Lemma
• Theorem 6
• Theorem 7
• Theorem 9
• Theorem 12