Monitoring graph edges via shortest paths: computational complexity and approximation algorithms

Edge-Geodetic Sets play a crucial role in network monitoring and optimization, wherein the goal is to strategically place monitoring stations on vertices of a network, represented as a graph, to ensure complete coverage of edges and mitigate faults by monitoring lines of communication. This paper illustrates and explores the Monitoring Edge-Geodetic Set (MEG-set) problem, which involves determining the minimum set of vertices that need to be monitored to achieve geodetic coverage for a given network. The significance of this problem lies in its potential to facilitate efficient network monitoring, enhancing the overall reliability and performance of various applications. In this work, we prove the $\mathcal{NP}$-completeness of the MEG-set optimization problem by showing a reduction from the well-known Vertex Cover problem. Furthermore, we present inapproximability results, proving that the MEG-set optimization problem is $\mathcal{APX}$-Hard and that, if the unique games conjecture holds, the problem is not approximable within a factor of $2-\epsilon$ for any constant $\epsilon > 0$. Despite its $\mathcal{NP}$-hardness, we propose an efficient approximation algorithm achieving an approximation ratio of $O(\sqrt{|V(G)| \cdot \ln{|V(G)|})}$ for the MEG-set optimization problem, based on the well-known Set Cover approximation algorithm, where $|V(G)|$ is the number of nodes of the MEG-set instance.