2024/06/29 by Bilò, Davide, Di Fonso, Alessia, Di Stefano, Gabriele +1 · 2 citations
#Computational Complexity (cs.CC) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2407.00409
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 μ-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 μ-set with size Ω( √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 ε > 0, we show that mutual-visibility and dual mutual-visibility problems are not approximable within a factor of n1/3-ε, while outer and total mutual-visibility problems are not approximable within a factor of n1/2 - ε, 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.