Guarding Polyominoes Under k-Hop Visibility

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 tha...

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Veröffentlicht in:Algorithmica Jg. 87; H. 4; S. 572 - 593
Hauptverfasser: Filtser, Omrit, Krohn, Erik, Nilsson, Bengt J., Rieck, Christian, Schmidt, Christiane
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 2025
Springer Nature B.V
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Apexes
Approximation
Art Gallery problem
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
hop dominating set
k-hop dominating set
k-hop visibility
Mathematics of Computing
Polyominoes
Shortest-path problems
Theory of Computation
VC dimension
Visibility
hop dominating set
VC dimension
Polyominoes
Approximation
Art Gallery problem
hop visibility
ISSN:0178-4617, 1432-0541, 1432-0541
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Zusammenfassung: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 .
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:0178-4617
1432-0541
1432-0541
DOI:10.1007/s00453-024-01292-7