Abstract
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.
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Ding, W., Xue, G. Minimum diameter cost-constrained Steiner trees. J Comb Optim 27, 32–48 (2014). https://doi.org/10.1007/s10878-013-9611-2
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DOI: https://doi.org/10.1007/s10878-013-9611-2