Cluster Before You Hallucinate: Approximating Node-Capacitated Network Design and Energy Efficient Routing

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 $O(\log² n)$ approximation to the cost with an $O(\log³ n)$ factor violation in node capacities. For multi-commodity demands, we obtain an $O(\log⁴ n)$ 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.