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arXiv · 1403.2972

Complexity of the General Chromatic Art Gallery Problem

Sándor P. Fekete·Stephan Friedrichs·Michael Hemmer
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Abstract

In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon $P$. We consider the Chromatic AGP (CAGP), where the guards are colored. As long as $P$ is completely covered, the number of guards does not matter, but guards with overlapping visibility regions must have different colors. This problem has applications in landmark-based mobile robot navigation: Guards are landmarks, which have to be distinguishable (hence the colors), and are used to encode motion primitives, \eg, "move towards the red landmark". Let $χ_G(P)$, the chromatic number of $P$, denote the minimum number of colors required to color any guard cover of $P$. We show that determining, whether $χ_G(P) \leq k$ is \NP-hard for all $k \geq 2$. Keeping the number of colors minimal is of great interest for robot navigation, because less types of landmarks lead to cheaper and more reliable recognition.

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Sándor P. Fekete, Stephan Friedrichs, Michael Hemmer. 2014-03-12. Complexity of the General Chromatic Art Gallery Problem. https://arxiv.org/abs/1403.2972

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