Maximizing the Guarded Interior of an Art Gallery

In the Art Gallery problem a polygon is given and the goal is to place as few guards as possible so that the entire area of the polygon is covered. We address a closely related problem: how to place a fixed number of guards on the vertices or the edges of a simple polygon so that the total guarded area inside the polygon is maximized. Recall that an optimization problem is called APX-hard, if there exists an > 0 such that an approximation ratio of 1+ cannot be guaranteed by any polynomial time approximation algorithm, unless P = NP . We prove that our problem is APX-hard and we present a polynomial time algorithm achieving constant approximation ratio. Finally we extend our results for the case where the guards are required to cover valued items inside the polygon. The valued items or “treasures” are modeled as simple closed polygons.