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Guarding Polyominoes Under k-Hop Visibility

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LATIN 2024: Theoretical Informatics (LATIN 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14578))

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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 \(\textsf{NP}\)-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 \(k\in \mathbb {N}\).

Due to space constraints, all missing details can be found in the full version [15].

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Funding

B. J. N. and C. S. are supported by grants 2021-03810 and 2018-04001 from the Swedish Research Council (Vetenskapsrådet). C. S. was supported by grant 2018-04101 from Sweden’s innovation agency VINNOVA.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, The Open University of Israel, Tel Aviv, Israel

    Omrit Filtser

  2. Department of Computer Science, University of Wisconsin - Oshkosh, Oshkosh, WI, USA

    Erik Krohn

  3. Department of Computer Science and Media Technology, Malmö University, Malmö, Sweden

    Bengt J. Nilsson

  4. Department of Computer Science, TU Braunschweig, Braunschweig, Germany

    Christian Rieck

  5. Department of Science and Technology, Linköping University, Norrköping, Sweden

    Christiane Schmidt

Authors
  1. Omrit Filtser
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  2. Erik Krohn
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  3. Bengt J. Nilsson
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  4. Christian Rieck
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  5. Christiane Schmidt
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Corresponding author

Correspondence to Christiane Schmidt .

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Editors and Affiliations

  1. DIM-CMM, Universidad de Chile, Santiago, Chile

    José A. Soto

  2. Technical University of Munich, Munich, Germany

    Andreas Wiese

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Filtser, O., Krohn, E., Nilsson, B.J., Rieck, C., Schmidt, C. (2024). Guarding Polyominoes Under k-Hop Visibility. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14578. Springer, Cham. https://doi.org/10.1007/978-3-031-55598-5_19

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  • DOI: https://doi.org/10.1007/978-3-031-55598-5_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-55597-8

  • Online ISBN: 978-3-031-55598-5

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