On the approximability of graph visibility problems

TL;DR

This work studies graph visibility problems through the four variants $\mu,\mu_{\rm o},\mu_{\rm d},\mu_{\rm t}$, formalizing mutual-visibility in graphs and its connections to the general position problem. It provides a polynomial-time algorithm achieving a $\mu$-set of size $|S|=\Omega\left(\sqrt{n/ Dbar}\right)$ via a 3-uniform hypergraph reduction and Caro–Wei bounds, and it establishes strong inapproximability: APX-hardness for all $\tau$ on diameter-2 graphs and tight bounds on inapproximability for diameter at least 3. The paper also analyzes the general-position number, showing a potentially large gap with the mutual-visibility number on diameter-2 graphs and proving no poly-time $n^{1-\varepsilon}$-approximation for gp. Together, these results delineate the computational limits of visibility-type graph invariants and stimulate questions about relaxed visibility concepts.

Abstract

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 $G$.

Figures (3)

• Figure 1: Examples of maximum $\tau$-sets on a graph $G$. Note that the $\mu_{\rm d}$-set is neither a $\mu_{\rm t}$-set nor a $\mu_{\rm o}$-set. The $\mu_{\rm d}$-set is neither a $\mu_{\rm t}$-set nor a $\mu_{\rm o}$-set. The $\mu_{\rm t}$-set is a feasible $\mu$-set, $\mu_{\rm d}$-set and $\mu_{\rm o}$-set that is not maximum.
• Figure 2: A sample graph $G$ and the hyperedges (shown as sets of vertices) added to the hypergraph $H$ for the pair of vertices $u,v$ when $\langle u, x_1, x_2, v\rangle$ is chosen as a shortest path.
• Figure 3: Construction of graph $G$ from graph $H$; $L=2$. Vertices inside the dashed ellipses induce complete graphs.

Theorems & Definitions (15)

• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8
• Corollary 9
• Theorem 10