AI 中文总结
研究具有凸可控处理时间的单机调度以最小化总加权完成时间问题，提出多项式时间$e \le 2.719$近似算法等，给出该问题近似性见解，证明简单排序规则对一般情况不能保证常数因子近似。
AI 中文摘要
我们研究了具有可控处理时间的单机调度问题，以最小化总加权完成时间，重点关注作业处理时间是其分配的连续资源的凸函数的情况。该问题的计算复杂性是一个长期存在的开放问题，目前既不知道它是否多项式时间可解，也不知道它是否是NP难的。虽然我们没有完全解决这个复杂性问题，但我们对该问题的近似性提供了一些见解。积极方面，我们提出了一种多项式时间的$e \le 2.719$近似算法，以及一种针对最大参数值由实例大小多项式界定的实例的拟多项式近似方案。消极方面，我们证明了文献中某些特殊情况下最优的简单排序规则不能保证一般情况下的常数因子近似。
英文摘要
We study the single-machine scheduling problem with controllable processing times to minimize the total weighted completion time, focusing on the setting where a job's processing time is a convex function of its allocated continuous resource. The computational complexity of this problem represents a long-standing open question, as it is currently neither known to be polynomial-time solvable nor NP-hard. While we do not fully resolve this complexity question, we provide several insights into the problem's approximability. On the positive side, we present a polynomial-time $e \le 2.719$-approximation algorithm, alongside a quasi-polynomial approximation scheme for instances where the largest parameter value is polynomially bounded by the instance size. On the negative side, we demonstrate that simple sorting rules, which are optimal for certain special cases in the literature, cannot guarantee a constant-factor approximation for the general case.
