A new algorithmic approach for solving the stochastic Steiner tree problem based on three procedures for computing lower bounds (dual ascent, Lagrangian relaxation, Benders decomposition) is introduced. Our method is derived from a new integer linear programming formulation, which is shown to be strongest among all known formulations. The resulting method, which relies on an interplay of the dual information retrieved from the respective dual procedures, computes upper and lower bounds and combines them with several rules for fixing variables in order to decrease the size of problem instances. The effectiveness of our method is compared in an extensive computational study with the state-of-the-art exact approach, which employs a Benders decomposition based on two-stage branch-and-cut, and a genetic algorithm introduced during the DIMACS implementation challenge on Steiner trees. Our results indicate that the presented method significantly outperforms existing ones, both on benchmark instances from literature, as well as on large-scale telecommunication networks.

中文翻译：

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两阶段随机斯坦纳树问题的分解方法。

介绍了一种基于计算下界的三个过程（对偶上升、拉格朗日松弛、Benders 分解）来解决随机 Steiner 树问题的新算法方法。我们的方法源自一种新的整数线性规划公式，该公式被证明是所有已知公式中最强的。由此产生的方法依赖于从各自的对偶过程中检索到的对偶信息的相互作用，计算上限和下限，并将它们与固定变量的多个规则相结合，以减少问题实例的大小。我们的方法的有效性在广泛的计算研究中与最先进的精确方法进行了比较，该方法采用基于两阶段分支和切割的 Benders 分解，以及在 DIMACS 实施挑战期间引入的遗传算法在施泰纳树上。我们的结果表明，所提出的方法无论是在文献基准实例上还是在大规模电信网络上都明显优于现有方法。