# Hallucination helps: Energy efficient virtual circuit routing

## Abstract

We consider virtual circuit routing protocols, with an objective of minimizing energy, in a network of components that are speed scalable, and that may be shutdown when idle. We assume that the speed s of a link is proportional to its load, and assume the standard model for component power, namely that the power is some constant static power a plus sα, where typically α ∈ [1.1,3]. We give a polynomial-time offline algorithm for multicommodity routing, that has approximation ratio 0(logαk), where k is the number of demand pairs. This is obtained as a combination of three natural combinatorial algorithms. The key step of the algorithm design is a random sampling technique that we call hallucination, which is reminiscent of the Sample-Augment framework for solving Buy-at-Bulk type problems, and sampling in cut-sparsification algorithms. The analysis of the approximation ratio is then a direct consequence of the flow-cut gap for multicommodity flow. The algorithm extends rather naturally to an online algorithm, which we show has competitive ratio Õ(log3α+1 k). The analysis of the online algorithm introduces a natural "priority" multicommodity flow problem, and bounds the priority multicommodity flow-cut gap- This might also be of independent interest. We also explain how our hallucination technique can be used to achieve an (0(log km), O(log km)) bicriteria approximation result for the problem of buying a minimum cost collection of unit- capacitated edges to support a concurrent multicommodity flow, where m is the number of links in the network. Copyright © 2014 by the Society for Industrial and Applied Mathematics.