Guarding Polyominoes Under k-Hop Visibility

Omrit Filtser; Erik Krohn; Bengt J Nilsson; Christian Rieck; Christiane Schmidt

doi:10.1007/s00453-024-01292-7

Guarding Polyominoes Under k-Hop Visibility

Omrit Filtser, Erik Krohn, Bengt J Nilsson, Christian Rieck, Christiane Schmidt

Computer Science

Research output: Contribution to journal › Article › peer-review

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 2×2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a 3×3 block of cells) for all k∈N.

Original language	English
Article number	4
Pages (from-to)	572-593
Number of pages	22
Journal	Algorithmica
Volume	87
Issue number	4
DOIs	https://doi.org/10.1007/s00453-024-01292-7
State	Published - 11 Jan 2025

Bibliographical note

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Keywords

Approximation
Art Gallery problem
k-hop dominating set
k-hop visibility
Polyominoes
VC dimension

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10.1007/s00453-024-01292-7

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title = "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 NP-complete even in thin polyominoes (i.e., polyominoes that do not a contain a 2×2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a 3×3 block of cells) for all k∈N.",

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AU - Krohn, Erik

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AU - Rieck, Christian

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N1 - Publisher Copyright: © The Author(s) 2025.

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N2 - 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 2×2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a 3×3 block of cells) for all k∈N.

AB - 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 2×2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a 3×3 block of cells) for all k∈N.

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KW - k-hop dominating set

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Guarding Polyominoes Under k-Hop Visibility

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