Minimum diameter cost‐constrained Steiner trees

Given an edge‐weighted undirected graph $G=(V,E,c,w)$ where each edge $e\in E$ has a cost $c(e)\ge 0$ and another weight $w(e)\ge 0$ , a set $S\subseteq V$ of terminals and a given constant ${\mathrm{C}}_0\ge 0$ , the aim is to find a minimum diameter Steiner tree whose all terminals appear as leaves and the cost of tree is bounded by ${\mathrm{C}}_0$ . The diameter of a tree refers to the maximum weight of the path connecting two different leaves in the tree. This problem is called the minimum diameter cost‐constrained Steiner tree problem, which is NP‐hard even when the topology of the Steiner tree is fixed. In this paper, we deal with the fixed‐topology restricted version. We prove the restricted version to be polynomially solvable when the topology is not part of the input and propose a weakly fully polynomial time approximation scheme (weakly FPTAS) when the topology is part of the input, which can find a $(1+\epsilon )$ –approximation of the restricted version problem for any $\epsilon >0$ with a specific characteristic.