Minimum Diameter K-Steiner Forest

Given an edge-weighted undirected graph G = (V, E, w) and a subset 𝒯⊆ V of p terminals, a k-Steiner forest spanning all the terminals in 𝒯 includes k branches, where every branch is a Steiner tree. The diameter of a k-Steiner forest is referred to as the maximum distance between two terminals of a branch. This paper studies the minimum diameter k-Steiner forest problem (MDkSFP) and establishes the relationship between MDkSFP and the absolute Steiner k-center problem (ASkCP). We first obtain a 2-approximation to ASkCP by a dual approximation algorithm and then achieve a 2-approximation to MDkSFP. Further, we achieve a (better) 2 ρ -approximation to MDkSFP, where ρ < 1 in general, by modifying the sites of centers and re-clustering all the terminals.