On the union of intermediate nodes of shortest paths

Abstract

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,v∈V M _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 $H(\frac{\delta (\delta -1)}{2})$-approximation for the problem where H(⋅) is the harmonic function.