Visibility Extension via Reflection

TL;DR

This paper explores how reflecting edges, especially diffuse reflections, can extend visibility in the Art Gallery problem, analyzing complexity and providing bounds on guard numbers with reflections.

Contribution

It introduces the impact of diffuse reflections on visibility and guard placement, extending previous work on specular reflections and NP-hardness in the Art Gallery problem.

Findings

Adding diffuse reflections reduces the number of guards needed by a factor related to reflection count.

Several visibility extension problems with reflections are NP-hard.

An approximation algorithm for guard placement is generalized to include reflections.

Abstract

This paper studies a variant of the Art Gallery problem in which the ``walls" can be replaced by \emph{reflecting edges}, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon $\cal P$, 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 $\pi$. Lee and Aggarwal already proved that several versions of the general Art Gallery problem are $NP$-hard. We show that several cases of adding an area to the visible area of a given point guard are $NP$-hard, too. Second, we assume 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 $r$ \emph{diffuse} reflections, the minimum number of \emph{vertex or boundary} guards required to cover a given simple polygon $\cal P$ decreases to { $\bf \lceil \frac{\alpha}{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}, where $\alpha$ indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the $\mathcal{O}(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.