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.

翻译：本文研究了艺术画廊问题的一个变体，其中“墙壁”可以被反射边缘替代，这使得守卫能够看得更远，从而覆盖画廊的更大区域。给定一个简单多边形 $\cal P$，首先，我们考虑一个点观察者作为守卫，并试图利用反射来为守卫的可见性多边形增加一定面积。我们研究了镜面反射和漫反射两种情况：镜面反射类似于镜像反射，而漫反射中入射光线与反射光线之间的夹角可以取 $0$ 到 $\pi$ 之间的任意值。Lee 和 Aggarwal 已证明一般艺术画廊问题的多个版本是 NP-难的。我们证明了在给定点守卫的可视区域中添加一块面积的若干情形也是 NP-难的。其次，我们假设所有边缘均为反射器，并旨在减少覆盖整个画廊所需的最小守卫数量。Chao Xu 证明，即使考虑 $r$ 次镜面反射，仍可能需要 $\lfloor \frac{n}{3} \rfloor$ 个守卫来覆盖多边形。设 $r$ 为守卫可见光线的最大反射次数。本文证明，在考虑 $r$ 次漫反射的情况下，覆盖给定简单多边形 $\cal P$ 所需的最小顶点或边界守卫数量减少至 { $\bf \lceil \frac{\alpha}{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}，其中 $\alpha$ 表示无反射时覆盖多边形所需的最小守卫数量。我们还将顶点守卫问题的 $\mathcal{O}(\log n)$-近似比算法推广至存在反射的场景中。