图可见性问题的可近似性研究
On the approximability of graph visibility problems
AI总结：
本文研究图互可见性及其变体的可近似性，提出多项式时间算法求得大小为 $Ω(\sqrt{n/\overline{D}})$ 的 $μ$-集，并给出不同直径图上的不可近似性下界，同时探讨了互可见性数与一般位置数的关系。
摘要：
可见性问题由于涉及具有挑战性的组合问题并与机器人导航问题相关，长期以来在不同假设下得到了广泛研究。在一个具有 $n$ 个顶点的图 $G$ 中，互可见性问题要求找到最大的顶点集合 $X\subseteq V(G)$（也称为 $μ$-集），使得对于任意两个顶点 $u,v\in X$，存在一条最短的 $u,v$-路径 $P$，且 $P$ 的所有内部顶点都不在 $X$ 中。这意味着 $u$ 和 $v$ 相对于 $X$ 是可见的。根据 $X$ 内部和/或外部顶点的可见性性质，该问题的变体被称为全互可见性、外互可见性和对偶互可见性问题。已知互可见性问题及其所有变体在直径为 $4$ 的图上都是 $\mathsf{NP}$-完全的。在本文中，我们设计了一个多项式时间算法，能够找到一个大小为 $Ω\left( \sqrt{n/ \overline{D}} \right)$ 的 $μ$-集，其中 $\overline D$ 是 $G$ 中任意两个顶点之间的平均距离。此外，我们给出了直径为 $2$ 的图上所有可见性问题的不可近似性结果，并加强了直径为 $3$ 或更大的图的不可近似性比率。更准确地说，对于直径至少为 $3$ 的图以及任意常数 $\varepsilon > 0$，我们证明除非 $\mathsf{P}=\mathsf{NP}$，否则互可见性和对偶互可见性问题不能在 $n^{1/3-\varepsilon}$ 的因子内被近似，而外互可见性和全互可见性问题不能在 $n^{1/2 - \varepsilon}$ 的因子内被近似。此外，我们研究了互可见性数与一般位置数之间的关系，其中 $X$ 中不存在三个不同的顶点 $u,v,w$ 同时属于 $G$ 的任何一条最短路径。
英文摘要：
Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph $G$ of $n$ vertices asks to find the largest set of vertices $X\subseteq V(G)$, also called $μ$-set, such that for any two vertices $u,v\in X$, there is a shortest $u,v$-path $P$ where all internal vertices of $P$ are not in $X$. This means that $u$ and $v$ are visible w.r.t. $X$. Variations of this problem are known as total, outer, and dual mutual-visibility problems, depending on the visibility property of vertices inside and/or outside $X$. The mutual-visibility problem and all its variations are known to be $\mathsf{NP}$-complete on graphs of diameter $4$. In this paper, we design a polynomial-time algorithm that finds a $μ$-set with size $Ω\left( \sqrt{n/ \overline{D}} \right)$, where $\overline D$ is the average distance between any two vertices of $G$. Moreover, we show inapproximability results for all visibility problems on graphs of diameter $2$ and strengthen the inapproximability ratios for graphs of diameter $3$ or larger. More precisely, for graphs of diameter at least $3$ and for every constant $\varepsilon > 0$, we show that mutual-visibility and dual mutual-visibility problems are not approximable within a factor of $n^{1/3-\varepsilon}$, while outer and total mutual-visibility problems are not approximable within a factor of $n^{1/2 - \varepsilon}$, unless $\mathsf{P}=\mathsf{NP}$. Furthermore we study the relationship between the mutual-visibility number and the general position number in which no three distinct vertices $u,v,w$ of $X$ belong to any shortest path of $G$.
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