We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines.

我们考虑将经典调度问题自然地概括为这样一种情况：在这种情况下，使用时间单位处理作业会导致一些与时间有关的成本，即使用时间费率，除了标准调度成本外，还必须支付该费用。我们专注于抢占式单机调度和两个经典调度成本函数，即（加权）完成时间和最大完成时间的总和，即工期。尽管这些问题在传统的调度环境中很容易解决，但在必须考虑使用时间费率的情况下，它们却要复杂得多。我们贡献了最佳的多项式时间算法和最佳的近似算法。对于将一台机器上的总（加权）完成时间减至最少的问题，我们提出了一种多项式时间算法，该算法为任何给定的作业序列计算最佳调度，即，用于根据给定序列抢先调度作业的最佳时隙集。该结果基于动态规划，其中使用了对最优解的结构和潜在函数自变量的精细分析。使用该算法，我们可以在多项式时间内最优地解决非加权问题。对于较普遍的问题，在这些问题中，作业可能具有不同的权重，我们基于引入的双重调度方法（用于在可变速度的机器上进行调度），开发了多项式时间近似方案（PTAS）。由于加权问题对NP的影响很大，因此我们的PTAS是我们所希望的最好的近似值。要进行抢先式调度以最大程度地减少制造时间，我们表明，存在一个具有多项式运行时间的相对简单的最优算法。即使在具有不相关机器的某种通用模型中，也是如此。