Complexity of the General Chromatic Art Gallery Problem

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

An art gallery approach to ensuring that landmarks are distinguishable

How many different classes of partially distinguishable landmarks are needed to ensure that a robot can always see a landmark without simultaneously seeing two of the same class? To study this, we introduce the chromatic art gallery problem. A guard set S ⊂ P is a set of points in a polygon P such that for all p ∈ P, there exists an s ∈ S such that s and p are mutually visible. Suppose that two members of a finite guard set S ⊂ P must be given different colors if their visible regions overlap. What is the minimum number of colors required to color any guard set (not necessarily a minimal guard set) of a polygon P? We call this number, χG(P), the chromatic guard number of P. We believe this problem has never been examined before, and it has potential applications to robotics, surveillance, sensor networks, and other areas. We show that for any spiral polygon Pspi, χG(Pspi) ≤ 2, and for any staircase polygon (strictly monotone orthogonal polygon) Psta, χG(Psta) ≤ 3. For lower bounds, we construct a polygon with 4k vertices that requires k colors. We also show that for any positive integer k, there exists a monotone polygon Mk with 3k2 vertices such that χG(Mk) ≥ k, and for any odd integer k, there exists an orthogonal polygon Rk with 4k2 + 10k + 10 vertices such that χG(Rk) ≥ k