Floodlight illumination of infinite wedges

The floodlight illumination problem asks whether there exists a one-to-one placement of n floodlights illuminating infinite wedges of angles α1, . . . , αn at n sites p1, . . . , pn in a plane such that a given infinite wedge W of angle θ located at point q is completely illuminated by the floodlights. We prove that this problem is NP-hard, closing an open problem from 2001 [6]. In fact, we show that the problem is NP-complete even when αi = α for all 1 ≤ i ≤ n (the uniform case) and θ = Pn i=1 αi (the tight case). On the positive side, we describe sufficient conditions on the sites of floodlights for which there are efficient algorithms to find an illumination. We discuss various approximate solutions and show that computing any finite approximation is NP-hard while ε-angle approximations can be obtained efficiently.