Single-commodity network design considers an edge-weighted, undirected graph with a supply/demand value at each node. It asks for minimum weight capacities such that each node can exactly send (or receive) its supply (or demand). In the robust variant, the supply or demand values may assume any realization in a given uncertainty set. One popular set is the well-known Hose polytope, which specifies an interval for the supply/demand at each node, while ensuring that the total supply and demand are balanced across the whole network. While previous work has established the Hose uncertainty set as a tractable choice, it can yield unnecessarily expensive solutions because it admits many unlikely supply and demand scenarios. In this paper, we propose a generalization of the Hose polytope that more realistically captures existing interdependencies among nodes in real life networks, and we show how to extend the state-of-the-art cutting plane algorithm for solving the single-commodity robust network design problem in view of this new uncertainty set. Our computational studies across multiple robust network design instances illustrate that the new set can provide significant cost savings without sacrificing numerical tractability.

中文翻译：

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单商品鲁棒网络设计中的广义软管不确定性

单商品网络设计考虑了一个边缘加权的无向图，在每个节点上都有供需值。它要求最小的权重容量，以便每个节点都可以准确地发送（或接收）其供应（或需求）。在稳健的变体中，供需值可以假定给定不确定性集中的任何实现。一种流行的设置是众所周知的Hose多拓扑，它指定每个节点的供求间隔，同时确保整个网络的总供需平衡。尽管先前的工作已将软管不确定性设置确定为易于处理的选择，但由于它接受了许多不太可能的供需情况，因此可以产生不必要的昂贵解决方案。在本文中，我们提出了一种软管多态性的概括，可以更实际地捕获现实网络中节点之间的现有相互依存关系，并且我们将展示如何扩展最新的切割平面算法以解决单商品鲁棒网络设计问题新的不确定性集合。我们对多个健壮的网络设计实例的计算研究表明，新的集合可以节省大量成本，而不会牺牲数值可处理性。