SIAM Journal on Computing, Volume 49, Issue 1, Page 37-66, January 2020.
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 shut down when idle. We assume the standard model for component power: the power consumed by a component with load (speed) $s$ is $\sigma+ s^\alpha$, where $\sigma$ is the static power and the exponent $\alpha>1$. We obtain a very simple $O(\log^\alpha k)$-approximation algorithm for multicommodity routing, where $k$ is the number of demand pairs. This improves upon previous results by several logarithmic factors. The key step in our algorithm is a random sampling technique that we call hallucination, which is reminiscent of the sample-augment framework for buy-at-bulk problems, and sampling in cut-sparsification algorithms. We also consider the online setting of the problem, where demand pairs arrive over time. We show that our offline algorithm naturally extends to the online setting, and obtain a randomized competitive ratio of $\tilde{O}( \log^{3\alpha + 1} k)$, which is the first nontrivial bound. The analysis of this algorithm involves the study of priority multicommodity flows, where edges and demand-pairs have priorities and each demand-pair must route its flow only on edges of lower priority. We establish a polylogarithmic flow-cut gap for these priority flows, which we believe is of independent interest. Finally, we show how our technique can be used to achieve a randomized $( O(\log m), O(\log^2 m))$ bicriteria competitive algorithm for the uniform capacitated network design problem, where $m$ is the number of edges. Here, every edge has a cost $c_e$ and uniform capacity $q$, and the goal is to choose the minimum cost subgraph that can support the given multicommodity demand. This is the first online algorithm for this problem. In fact, our approach also improves prior results in the offline setting by several logarithmic factors.

中文翻译：

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幻觉帮助：节能虚拟电路布线

SIAM计算杂志，第49卷，第1期，第37-66页，2020年1月。
我们考虑虚拟电路路由协议，其目的是使速度可伸缩的组件网络中的能量最小化，并且在空闲时可以将其关闭。我们假设组件功率为标准模型：负载（速度）为$ s $的组件所消耗的功率为$ \ sigma + s ^ \ alpha $，其中$ \ sigma $是静态功率，指数$ \ alpha> 1 $。对于多商品路由，我们获得了一个非常简单的$ O（\ log ^ \ alpha k）$近似算法，其中$ k $是需求对的数量。这通过几个对数因子改善了先前的结果。我们算法的关键步骤是一种称为幻觉的随机采样技术，这种技术使人联想到批量购买问题的样本增广框架以及采用可分割化算法进行采样。我们还考虑了问题的在线设置，需求对随时间到达的地方。我们证明了我们的离线算法自然会扩展到在线设置，并获得$ \ tilde {O}（\ log ^ {3 \ alpha + 1} k）$的随机竞争比率，这是第一个非平凡的界限。该算法的分析涉及优先级多商品流的研究，其中边缘和需求对具有优先级，每个需求对必须仅在较低优先级的边缘路由其流。我们为这些优先流程建立了一个多对数流量削减缺口，我们认为这是独立利益。最后，我们展示了如何针对统一电容网络设计问题使用我们的技术来实现随机的$（O（\ log m），O（\ log ^ 2 m））$双标准竞争算法，其中$ m $是边数。在这里，每个边的成本为$ c_e $，容量为qq $，目标是选择能够支持给定的多商品需求的最小成本子图。这是解决此问题的第一个在线算法。实际上，我们的方法还通过几个对数因素改善了脱机设置中的先前结果。