On the Inapproximability of Finding Minimum Monitoring Edge-Geodetic Sets

Given an undirected connected graph G = (V(G), E(G)) on n vertices, the minimum Monitoring Edge-Geodetic Set (MEG-set) problem asks to find a subset M ⊆ V(G) of minimum cardinality such that, for every edge e ∈ E(G), there exist x,y ∈ M for which all shortest paths between x and y in G traverse e. We show that, for any constant c < 1/2, no polynomial-time (c log n)-approximation algorithm for the minimum MEG-set problem exists, unless 𝖯 = 𝖭𝖯.