We consider the problem of energy-efficient scheduling across multiple processors with a power-down mechanism. In this setting a set of $n$ jobs with individual release times, deadlines, and processing volumes must be scheduled across $m$ parallel processors while minimizing the consumed energy. Idle processors can be turned off to save energy, while turning them on requires a fixed amount of energy. For the special case of a single processor, the greedy Left-to-Right algorithm guarantees an approximation factor of $2$. We generalize this simple greedy policy to the case of $m \geq 1$ processors running in parallel and show that the energy costs are still bounded by $2 \text{OPT} + P$, where $\text{OPT}$ is the energy consumed by an optimal solution and $P < \text{OPT}$ is the total processing volume. Our algorithm has a running time of $\mathcal{O}(n f \log d)$, where $d$ is the difference between the latest deadline and the earliest release time, and $f$ is the running time of a maximum flow calculation in a network of $\mathcal{O}(n)$ nodes.

翻译：我们考虑在具有功率关闭机制的多处理器上进行能量高效调度的问题。在此设置中，一组 $n$ 个具有各自释放时间、截止时间和处理量的作业需要在 $m$ 个并行处理器上调度，同时最小化消耗的能量。空闲处理器可关闭以节省能量，而开启它们需要固定量的能量。对于单处理器的特例，贪心从左到右算法保证了 $2$ 的近似因子。我们将这种简单的贪心策略推广到 $m \geq 1$ 个处理器并行运行的情况，并表明能量成本仍然受 $2 \text{OPT} + P$ 的限制，其中 $\text{OPT}$ 是最优解消耗的能量，$P < \text{OPT}$ 是总处理量。我们的算法运行时间为 $\mathcal{O}(n f \log d)$，其中 $d$ 是最晚截止时间与最早释放时间之差，$f$ 是在 $\mathcal{O}(n)$ 个节点的网络中计算最大流的运行时间。