  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].
