Face-Guarding Polyhedra

We study the Art Gallery Problem for face guards in polyhedral environments. The problem can be informally stated as: how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate it? We consider both closed and open face guards (i.e., faces with or without their boundary), and we give some upper and lower bounds on the minimum number of faces required to guard a given polyhedron, in terms of the total number of its faces, f . In some notable cases, our bounds are tight: bf/6c open face guards for orthogonal polyhedra, and bf/4c open face guards for 4-oriented polyhedra (i.e., polyhedra whose faces have only four different orientations). Then we show that it is NP-hard to approximate the minimum number of (closed or open) face guards within a factor of Ω(log f), even for polyhedra that are orthogonal and simply connected. Along the way we discuss some applications, arguing that face guards are not a reasonable model for guards patrolling on the surface of a polyhedron.