On conflict-free chromatic guarding of simple polygons

Abstract

We study the problem of colouring the vertices of a polygon, such that every viewer in it can see a unique colour. The goal is to minimise the number of colours used. This is also known as the conflict-free chromatic guarding problem with vertex guards, and is motivated, e.g., by the problem of radio frequency assignment to sensors placed at the polygon vertices. We first study the scenario in which viewers can be all points of the polygon (such as a mobile robot which moves in the interior of the polygon). We efficiently solve the related problem of minimising the number of guards and approximate (up to only an additive error) the number of colours required in the special case of polygons called funnels. Then we give an upper bound of O(log^2 n) colours on n-vertex weak visibility polygons, by decomposing the problem into sub-funnels. This bound generalises to all simple polygons. We briefly look also at the second scenario, in which the viewers are only the vertices of the polygon. We show a lower bound of 3 colours in the general case of simple polygons and conjecture that this is tight. We also prove that already deciding whether 1 or 2 colours are enough is NP-complete.