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Vertex-to-Point Conflict-Free Chromatic Guarding is NP-Hard

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WALCOM: Algorithms and Computation (WALCOM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13174))

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Abstract

The art gallery problem is to find a set of guards who together can observe every point of the interior of a polygon P. We study a chromatic variant of the problem, where each guard is assigned one of k distinct colors. A chromatic guarding is said to be conflict-free if at least one of the colors seen by every point in P is unique (i.e., each point in P is seen by some guard whose color appears exactly once among the guards visible to that point). In this paper, we consider vertex-to-point guarding, where the guards are placed on vertices of P, and they observe every point of the interior of P. The vertex-to-point conflict-free chromatic art gallery problem is to find a colored-guard set such that (i) guards are placed on P’s vertices, and (ii) any point in P can see a guard of a unique color among all the visible guards. In this paper, it is shown that determining whether there exists a conflict-free chromatic vertex-guard set for a polygon with holes is NP-hard when the number of colors is \(k=2\).

This work was supported by JSPS KAKENHI Grant Number 16K00020.

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Authors and Affiliations

  1. Hiroshima University, Higashi-Hiroshima, 739-8521, Japan

    Chuzo Iwamoto & Tatsuaki Ibusuki

Authors
  1. Chuzo Iwamoto
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  2. Tatsuaki Ibusuki
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Correspondence to Chuzo Iwamoto .

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Editors and Affiliations

  1. University of Bonn, Bonn, Germany

    Petra Mutzel

  2. Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh

    Md. Saidur Rahman

  3. Universitas Jember, Jember, Indonesia

    Slamin

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Iwamoto, C., Ibusuki, T. (2022). Vertex-to-Point Conflict-Free Chromatic Guarding is NP-Hard. In: Mutzel, P., Rahman, M.S., Slamin (eds) WALCOM: Algorithms and Computation. WALCOM 2022. Lecture Notes in Computer Science(), vol 13174. Springer, Cham. https://doi.org/10.1007/978-3-030-96731-4_10

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  • DOI: https://doi.org/10.1007/978-3-030-96731-4_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-96730-7

  • Online ISBN: 978-3-030-96731-4

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