Complexity of the General Chromatic Art Gallery Problem

TL;DR

This paper investigates the complexity of the Chromatic Art Gallery Problem, proving that determining the minimum number of colors needed for guard coverage in polygons is NP-hard for all cases where two or more colors are used.

Contribution

It establishes the NP-hardness of the Chromatic AGP for any number of colors greater than or equal to two, highlighting the computational difficulty of minimal landmark coloring.

Findings

Determining if _G(P) ; k is NP-hard for all k ; 2

Minimal coloring is computationally challenging for robot navigation applications

Highlights the complexity of landmark distinguishability in polygon coverage

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 $\chi_G(P)$, the chromatic number of $P$, denote the minimum number of colors required to color any guard cover of $P$. We show that determining, whether $\chi_G(P) \leq k$ 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.