This paper studies a variant of the Art-gallery problem in which "walls" can be replaced by \emph{reflecting-edges}, which allows the guard to see further and thereby see a larger portion of the gallery. The art-gallery is a simple closed polygon $P$, a guard is a point $p$ in $P$, and a guard sees another point $q$ in $P$ if the segment $\overline{pq}$ is contained in the interior of $P$. The visibility region of $p$ is the set of points $q$ in $P$ that are visible from $p$. If we let an edge of the polygon allow reflections, then the visibility region should be changed accordingly. We study visibility with specular and diffuse reflections. Moreover, the number of times a ray can be reflected can be taken as a parameter. For \emph{vertex} guarding polygons with $k$ diffuse reflections, we establish an upper bound on the optimum solution. For this problem, we generalize the $O(logn)$-approximation ratio algorithm of the Art Gallery Problem. For a bounded $k$, the generalization gives a polynomial-time algorithm with $O(log n)$-approximation ratio for both cases diffuse and specular reflections. Furthermore, We show that several cases of the generalized problem are NP-hard. We also illustrate that if $P$ is a funnel or a weak visibility polygon, then the problem becomes more straightforward and can be solved in polynomial time.

中文翻译：

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通过反射扩展可见性

本文研究了艺术画廊问题的一个变体，其中“墙”可以用 \emph {reflecting-edges} 代替，这允许守卫看得更远，从而看到画廊的更大部分。画廊是一个简单的封闭多边形 $P$，守卫是 $P$ 中的一个点 $p$，如果段 $\overline{pq}$ 是一个守卫看到 $P$ 中的另一个点 $q$包含在 $P$ 的内部。$p$ 的可见区域是$P$ 中从$p$ 可见的点$q$ 的集合。如果我们让多边形的边缘允许反射，那么可见区域应该相应地改变。我们研究镜面反射和漫反射的可见性。此外，可以将光线反射的次数作为参数。对于 \emph{vertex} 保护具有 $k$ 漫反射的多边形，我们建立了最优解的上限。对于这个问题，我们概括了 Art Gallery Problem 的 $O(logn)$-approximation ratio 算法。对于有界的 $k$，泛化给出了一个多项式时间算法，对于漫反射和镜面反射这两种情况，都有 $O(log n)$-近似比。此外，我们证明了广义问题的几种情况是 NP 难的。我们还说明，如果 $P$ 是漏斗或弱可见性多边形，那么问题就变得更加简单，可以在多项式时间内解决。我们证明了广义问题的几种情况是 NP 难的。我们还说明，如果 $P$ 是漏斗或弱可见性多边形，那么问题就变得更加简单，可以在多项式时间内解决。我们证明了广义问题的几种情况是 NP 难的。我们还说明，如果 $P$ 是漏斗或弱可见性多边形，那么问题就变得更加简单，可以在多项式时间内解决。