While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints which might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the $\ell_p$-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the $\ell_{\infty}$-norm) and the number of edges (the $\ell_1$-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogrithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Kortsarz and Nutov, Discret. Appl. Math. 2022] and [Guo et al., Algorithmica 2022].

翻译：尽管网络设计大多聚焦于成本（边的数量或权重），节点度数也始终扮演着重要角色。传统上，度数要么作为目标函数以最小化最大度数（例如最小度数生成树问题），要么作为可能被违反的约束条件以提供双准则逼近（例如最小成本有界度数生成树问题）。我们从两个方向拓展了网络设计中的度数研究。首先，我们引入并研究了一种幸存网络设计问题的新变体，其中除了最小化选定边成本的传统目标外，我们还增加了一个约束条件：节点度向量的$\ell_p$-范数需受限于输入参数。这插值了最大度数（$\ell_{\infty}$-范数）与边数（$\ell_1$-度数）之间的经典情形，并在分布式系统和超大规模集成电路设计中具有自然应用。我们通过凸规划方法给出了两个度量标准下的常数双准则逼近。其次，我们为有界树宽图上的有界度数群斯坦纳问题提供了多项对数双准则逼近，这解决了[Kortsarz and Nutov, Discret. Appl. Math. 2022]与[Guo et al., Algorithmica 2022]中提出的一个开放问题。