We consider the following node-capacitated network design problem. The input is an undirected graph, set of demands, uniform node capacity and arbitrary node costs. The goal is to find a minimum node-cost subgraph that supports all demands concurrently subject to the node capacities. We consider both single and multi-commodity demands, and provide the first poly-logarithmic approximation guarantees. For single-commodity demands (i.e., all request pairs have the same sink node), we obtain an $O(\log^2 n)$ approximation to the cost with an $O(\log^3 n)$ factor violation in node capacities. For multi-commodity demands, we obtain an $O(\log^4 n)$ approximation to the cost with an $O(\log^{10} n)$ factor violation in node capacities. We use a variety of techniques, including single-sink confluent flows, low-load set cover, random sampling and cut-sparsification. We also develop new techniques for clustering multicommodity demands into (nearly) node-disjoint clusters, which may be of independent interest. Moreover, this network design problem has applications to energy-efficient virtual circuit routing. In this setting, there is a network of routers that are speed scalable, and that may be shutdown when idle. We assume the standard model for power: the power consumed by a router with load (speed) $s$ is $\sigma + s^\alpha$ where $\sigma$ is the static power and the exponent $\alpha > 1$. We obtain the first poly-logarithmic approximation algorithms for this problem when speed-scaling occurs on nodes of a network.

翻译：我们考虑以下节点容量网络设计问题。输入为一个无向图、一组需求、统一的节点容量和任意节点成本。目标是在满足节点容量的前提下，找到一个能同时支持所有需求的最小节点成本子图。我们同时考虑单商品和多商品需求，并首次给出了多项式对数近似保证。对于单商品需求（即所有请求对具有相同汇节点），我们实现了成本上的$O(\log^2 n)$近似，节点容量违反因子为$O(\log^3 n)$。对于多商品需求，我们实现了成本上的$O(\log^4 n)$近似，节点容量违反因子为$O(\log^{10} n)$。我们采用了多种技术，包括单汇合流、低负载集合覆盖、随机采样和割稀疏化。我们还开发了将多商品需求聚类成（近似）节点不相交簇的新技术，这可能具有独立的研究价值。此外，该网络设计问题可应用于能效虚拟电路路由。在此场景中，存在一个由速度可扩展且空闲时可关闭的路由器组成的网络。我们采用标准功率模型：负载（速度）为$s$的路由器消耗的功率为$\sigma + s^\alpha$，其中$\sigma$为静态功率，指数$\alpha > 1$。我们首次给出了当节点速度可扩展时该问题的多项式对数近似算法。