Hidden Mobile Guards in Simple Polygons

We consider guarding classes of simple polygons using mobile guards (polygon edges and diagonals) under the constraint that no two guards may see each other. In contrast to most other art gallery problems, existence is the primary question: does a specific type of polygon admit some guard set? Types include simple polygons and the subclasses of orthogonal, monotone, and starshaped polygons. Additionally, guards may either exclude or include the endpoints (so-called open and closed guards). We provide a nearly complete set of answers to existence questions of open and closed edge, diagonal, and mobile guards in simple, orthogonal, monotone, and starshaped polygons, with some surprising results. For instance, every monotone or starshaped polygon can be guarded using hidden open mobile (edge or diagonal) guards, but not necessarily with hidden open edge or hidden open diagonal guards.Comment: An abstract (6-page) version of this paper is to appear in the proceedings of CCCG 201

Minimum Hidden Guarding of Histogram Polygons

A hidden guard set $G$ is a set of point guards in polygon $P$ that all points of the polygon are visible from some guards in $G$ under the constraint that no two guards may see each other. In this paper, we consider the problem for finding minimum hidden guard sets in histogram polygons under orthogonal visibility. Two points $p$ and $q$ are orthogonally visible if the orthogonal bounding rectangle for $p$ and $q$ lies within $P$. It is known that the problem is NP-hard for simple polygon with general visibility and it is true for simple orthogonal polygon. We proposed a linear time exact algorithm for finding minimum hidden guard set in histogram polygons under orthogonal visibility. In our algorithm, it is allowed that guards place everywhere in the polygon

Optimally Guarding 2-Reflex Orthogonal Polyhedra by Reflex Edge Guards

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.Comment: 35 pages, 16 figure