On the approximability of graph visibility problems

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 V (G), also called -set, such that for any two vertices u, v 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 NP-complete on graphs of diameter 4. In this paper, we design a polynomial-time algorithm that finds a -set with size (n/ D), where 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 > 0, we show that mutual-visibility and dual mutual-visibility problems are not approximable within a factor of n^1/3-, while outer and total mutual-visibility problems are not approximable within a factor of n^1/2 -, unless P=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.