How many landmark colors are needed to avoid confusion in a polygon?

摘要

Suppose that two members of a finite point guard set S within a polygon P must be given different colors if their visible regions overlap, and that every point in P is visible from some point in S. The chromatic art gallery problem, introduced in [7], asks for the minimum number of colors required to color any guard set (not necessarily a minimal guard set) of P. We study two related problems. First, given a polygon P and a guard set S of P, can the members of S be efficiently and optimally colored so that no two members of S that have overlapping visibility regions have the same color? Second, given a polygon P and a set of candidate guard locations N, is it possible to efficiently and optimally choose the guard set S ⊆ N that requires the minimum number of colors? We provide an algorithm that solves the first question in polynomial time, and demonstrate the NP-hardness of the second question. Both questions are motivated by common robot tasks such as mapping and surveillance.