Visibility extension via reflection

Abstract

This paper studies a variant of the Art Gallery problem in which the “walls” can be replaced by reflecting edges, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon $\text{P}$, first, we consider one guard as a point viewer, and we intend to use reflection to add a certain amount of area to the visibility polygon of the guard. We study visibility with specular and diffuse reflections where the specular type of reflection is the mirror-like reflection, and in the diffuse type of reflection, the angle between the incident and reflected ray may assume all possible values between 0 and π. Lee and Aggarwal already proved that several versions of the general Art Gallery problem are $\text{NP}$-hard. We show that several cases of adding an area to the visible area of a given point guard are $\text{NP}$-hard, too.

Second,¹ we assume that all edges are reflectors, and we intend to decrease the minimum number of guards required to cover the whole gallery.

Chao Xu proved that even considering r specular reflections, one may need $\lfloor \frac{n}{3}\rfloor$ guards to cover the polygon, let r be the maximum number of reflections of a guard's visibility ray.

In this work, we prove that considering r diffuse reflections, the minimum number of vertex or boundary guards required to cover a given simple polygon $P$ decreases to $\lceil \frac{\alpha }{1+\lfloor \frac{r}{8}\rfloor }\rceil$, where α indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the $O(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.