Before revision:  We efficiently solve the related problem of minimizing the number of guards and approximate (up to only an additive error) the number of colours  in the special case of funnels. We also give an upper bound of O(log^2 n) colours on weak-visibility polygonswhich generalizes to all simple polygons.
After revision:  We efficiently solve the related problem of minimizing the number of guards and approximate (up to only an additive error) the number of colours in the special case of polygons called funnels. Then we give an upper bound of O(log^2 n) colours on weak-visibility polygonswhich generalizes to all simple polygons.