In this paper, we describe new complexity results and approximation algorithms for single-machine scheduling problems with non-renewable resource constraints and the total weighted completion time objective. This problem is hardly studied in the literature. Beyond some complexity results, only a fully polynomial-time approximation scheme (FPTAS) is known for a special case. In this paper, we discuss some polynomially solvable special cases and also show that under very strong assumptions, such as the processing time, the resource consumption and the weight is the same for each job; minimizing the total weighted completion time is still NP-hard. In addition, we also propose a 2-approximation algorithm for this variant and a polynomial-time approximation scheme (PTAS) for the case when the processing time equals the weight for each job, while the resource consumptions are arbitrary.

中文翻译：

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在受不可再生资源限制的单个机器上最小化总加权完成时间

在本文中，我们描述了具有不可再生资源约束和总加权完成时间目标的单机调度问题的新复杂性结果和近似算法。这个问题在文献中几乎没有研究过。除了某些复杂性结果之外，对于特殊情况，只有完全多项式时间近似方案 (FPTAS) 是已知的。在本文中，我们讨论了一些多项式可解的特殊情况，并表明在非常强的假设下，例如每个作业的处理时间、资源消耗和权重是相同的；最小化总加权完成时间仍然是 NP-hard。此外，我们还针对该变体提出了一种 2-近似算法，并针对处理时间等于每个作业的权重的情况提出了多项式时间近似方案 (PTAS)，