We study the computational complexity of scheduling jobs on a single speed-scalable processor with the objective of capturing the trade-off between the (weighted) flow time and the energy consumption. This trade-off has been extensively explored in the literature through a number of problem formulations that differ in the specific job characteristics and the precise objective function. Nevertheless, the computational complexity of four important problem variants has remained unresolved and was explicitly identified as an open question in prior work. In this paper, we settle the complexity of these variants. More specifically, we prove that the problem of minimizing the objective of total (weighted) flow time plus energy is NP-hard for the cases of (i) unit-weight jobs with arbitrary sizes, and (ii)~arbitrary-weight jobs with unit sizes. These results extend to the objective of minimizing the total (weighted) flow time subject to an energy budget and hold even when the schedule is required to adhere to a given priority ordering. In contrast, we show that when a completion-time ordering is provided, the same problem variants become polynomial-time solvable. The latter result highlights the subtle differences between priority and completion orderings for the problem.

翻译：本研究探讨了在单个速度可调处理器上调度作业的计算复杂度，旨在刻画（加权）流时间与能耗之间的权衡关系。该权衡关系在现有文献中已通过多种问题形式进行了广泛探索，这些形式在具体作业特征和精确目标函数方面存在差异。然而，四个重要问题变体的计算复杂度仍未得到解决，并在先前工作中被明确列为开放性问题。本文中，我们确定了这些变体的复杂度。具体而言，我们证明了在以下两种情况下最小化总（加权）流时间与能耗之和的目标函数是NP难的：（i）具有任意大小的单位权重作业，以及（ii）具有单位大小的任意权重作业。这些结果可推广至在能量预算约束下最小化总（加权）流时间的目标函数，且即使调度方案需遵循给定的优先级排序时仍然成立。相比之下，我们证明当提供完成时间排序时，相同的问题变体可在多项式时间内求解。后一结果凸显了该问题中优先级排序与完成时间排序之间的微妙差异。