On the union of intermediate nodes of shortest paths

Consider a connected graph G=(V,E). For a pair of nodes u and v, denote by M_uv the set of intermediate nodes of a shortest path between u and v. We are intertested in minimization of the union ∪ _u,vεVM_uv . We will show that this problem is NP-hard and cannot have polynomial-time ρlnδ-approximation for 0<ρ<1 unless NP ⊆ DTIME(n ^O(loglogn)) where δ is the maximum node degree of input graph. We will also construct a polynomial-time H(Δ(Δ-1)/2) -approximation for the problem where H(·) is the harmonic function.