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 $\mu$-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 $\mu$-set with size $\Omega\left( \sqrt{n/ \overline{D}} \right)$, 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 $G$.

翻译：可见性问题在不同假设下已被研究很长时间，因为它们构成了具有挑战性的组合问题，并且与机器人导航问题相关联。图$G$（具有$n$个顶点）中的互可见性问题要求找到最大的顶点子集$X\subseteq V(G)$（也称为$\mu$-集），使得对于任意两个顶点$u,v\in X$，存在一条最短$u,v$-路径$P$，且$P$的所有内部顶点都不在$X$中。这意味着$u$和$v$关于$X$是相互可见的。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$4$的图上是$\mathsf{NP}$-完全的。本文设计了一种多项式时间算法，该算法可以找到一个规模为$\Omega\left( \sqrt{n/ \overline{D}} \right)$的$\mu$-集，其中$\overline D$是$G$中任意两个顶点之间的平均距离。此外，我们展示了直径$2$的图上所有可见性问题的不可近似性结果，并加强了直径$3$或更大图上不可近似性的比率。更准确地说，对于直径至少为$3$的图以及每个常数$\varepsilon > 0$，我们证明互可见性和对偶互可见性问题不能在$n^{1/3-\varepsilon}$因子内近似，而外互可见性和全互可见性问题不能在$n^{1/2 - \varepsilon}$因子内近似，除非$\mathsf{P}=\mathsf{NP}$。此外，我们研究了互可见数（mutual-visibility number）与一般位置数（general position number）之间的关系，后者要求$X$中任意三个不同顶点$u,v,w$不属于$G$的任何最短路径。