Algorithms for the minimum diameter terminal Steiner tree problem

Given an undirected connected graph $$G=(V(G),E(G),d)$$ with a function $$d(\cdot )\ge 0$$ on edges and a subset $$S\subseteq V(G)$$ of terminals, the minimum diameter terminal Steiner tree problem (MDTSTP) asks for a terminal Steiner tree in $$G$$ of a minimum diameter. In the paper, the diameter of a tree refers to the longest of all the distances between two different leaves of the tree. When $$G$$ is a complete graph and $$d(\cdot )$$ is a metric function, we demonstrate that an optimal solution of MDTSTP is monopolar or dipolar and give an $$O(|S|\cdot |V(G)\setminus S|^2)$$-time exact algorithm. For the nonmetric version of MDTSTP, we present a simple 2-approximation algorithm with a time complexity of $$O(|V(G)\setminus S|\log |S|)$$, as well as two exact algorithms with a time complexity of $$O(|S|^3|V(G)|^2)$$ and $$O(|S|\cdot |V(G)\setminus S|^2+|S|^2\cdot |V(G)\setminus S|)$$, respectively.