通过反射扩展可见性
Visibility Extension via Reflection
AI总结：
本文研究Art Gallery问题的反射变体，证明增加点守卫可见区域的相关问题是NP-hard的，并推导出考虑漫反射时覆盖多边形所需顶点守卫最小数量的数学下界，同时推广了近似算法。
摘要：
本文研究了 Art Gallery 问题的一个变体，其中“墙壁”可以被替换为反射边，这使得守卫能够看得更远，从而看到画廊的更大区域。给定一个简单多边形 $\cal P$，首先，我们将一个守卫视为一个点观察者，并打算使用反射来增加该守卫可见性多边形的一定面积。我们研究了镜面反射和漫反射下的可见性，其中镜面反射类型是类似镜子的反射，而在漫反射类型中，入射光线和反射光线之间的角度可以取 $0$ 到 $\pi$ 之间的所有可能值。Lee 和 Aggarwal 已经证明了一般 Art Gallery 问题的几个版本是 $NP$-hard 的。我们证明了在给定一个点守卫的可见区域中增加面积的几个情况也是 $NP$-hard 的。其次，我们假设所有边都是反射器，并打算减少覆盖整个画廊所需的守卫的最小数量。Chao Xu 证明了即使考虑 $r$ 次镜面反射，也可能需要 $\lfloor \frac{n}{3} \rfloor$ 个守卫来覆盖该多边形。设 $r$ 为守卫可见性光线的最大反射次数。在本研究中，我们证明了考虑 $r$ 次漫反射时，覆盖给定简单多边形 $\cal P$ 所需的顶点或边界守卫的最小数量减少至 {$\bf \lceil \fracα{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}，其中 $α$ 表示在没有反射的情况下覆盖该多边形所需的最小守卫数量。我们还将顶点守卫问题的 $\mathcal{O}(\log n)$-近似比算法推广到存在反射的情况下使用。
英文摘要：
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 $π$. 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 { $\bf \lceil \fracα{1+ \lfloor \frac{r}{8} \rfloor} \rceil$}, where $α$ indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the $\mathcal{O}(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.
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