Optimally Guarding 2-Reflex Orthogonal Polyhedra by Reflex Edge Guards

Abstract: Let an orthogonal polyhedron be the union of a finite set of boxes in $\mathbb R^3$ (i.e., cuboids with edges parallel to the coordinate axes), whose surface is a connected 2-manifold. We study the NP-complete problem of guarding a non-convex orthogonal polyhedron having reflex edges in just two directions (as opposed to three, in the general case) by placing the minimum number of edge guards on reflex edges only. We show that $$\left\lfloor \frac{r-g}{2} \right\rfloor +1$$ reflex edge guards are sufficient, where $r$ is the number of reflex edges and $g$ is the polyhedron's genus. This bound is tight for $g=0$. We thereby generalize a classic planar Art Gallery theorem of O'Rourke, which states that the same upper bound holds for vertex guards in an orthogonal polygon with $r$ reflex vertices and $g$ holes. Then we give a similar upper bound in terms of $m$, the total number of edges in the polyhedron. We prove that $$\left\lfloor \frac{m-4}{8} \right\rfloor +g$$ reflex edge guards are sufficient, whereas the previous best known bound was $\lfloor 11m/72+g/6\rfloor-1$ edge guards (not necessarily reflex). We also consider the setting in which guards are open (i.e., they are segments without the endpoints), proving that the same results hold even in this more challenging case. Finally, we show how to compute guard locations matching the above bounds in $O(n \log n)$ time.