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.