Minimum diameter cost-constrained Steiner trees

Given an edge-weighted undirected graph G=(V,E,c,w) where each edge eεE has a cost c(e)≥0 and another weight w(e)≥0, a set S⊂ V of terminals and a given constant \mathrm{C}-0≥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+ε) -approximation of the restricted version problem for any ε >0 with a specific characteristic.