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

TL;DR

This work studies node-capacitated network design problems (MCNC and SSNC) on undirected graphs with uniform capacity $q$ and node costs $c_v$, aiming to minimize cost while supporting all demands concurrently. It introduces a two-tier approach: first obtain poly-logarithmic bicriteria approximations for SSNC using confluent flows and clustering, then extend to MCNC via iterative clustering into heavy/internal clusters and Hallucination-based routing across clusters with cut-sparsification, achieving poly-log performance with controlled node congestion. The results include an $(O(\log^{2} n), O(\log^{3} n))$ bicriteria for SSNC and an $(O(\log^{2} n \log^{2} k), O(\log^{6} n \log^{4} k))$ bicriteria for MCNC, further translating to energy-efficient routing guarantees via the NEERP reduction. These contributions advance the understanding of capacitated network design and provide practical poly-log algorithms for energy-aware routing in speed-scalable, node-centric networks.

Abstract

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 $σ+ s^α$ where $σ$ is the static power and the exponent $α> 1$. We obtain the first poly-logarithmic approximation algorithms for this problem when speed-scaling occurs on nodes of a network.

Figures (4)

• Figure 1: Graph $G_1$ for t1-active clusters.
• Figure 2: Graph $G_2$ for t2-active clusters.
• Figure 3: Different types of clusters at the end of an $\mathsf{MCcluster}$ iteration. The edges represent request-pairs (in $K$) across clusters.
• Figure 4: Un-contracting hallucinated graph $H_C$ using clusters ${\cal T}_h\xspace$ and flow ${\cal H}$.

Theorems & Definitions (34)

• Theorem 1: AndrewsAZ10
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 5
• Definition 6: SSNC Cluster
• Theorem 7: Theorem 20 in ChenKLRSV07
• Lemma 8
• Lemma 9
• Theorem 10: BabenkoGGN12