In moldable job scheduling, we are provided $m$ identical machines and $n$ jobs that can be executed on a variable number of machines. The execution time of each job depends on the number of machines assigned to execute that job. For the specific problem of monotone moldable job scheduling, jobs are assumed to have a processing time that is non-increasing in the number of machines. The previous best-known algorithms are: (1) a polynomial-time approximation scheme with time complexity $Ω(n^{g(1/\varepsilon)})$, where $g(\cdot)$ is a super-exponential function [Jansen and Thöle '08; Jansen and Land '18], (2) a fully polynomial approximation scheme for the case of $m \geq 8\frac{n}{\varepsilon}$ [Jansen and Land '18], and (3) a $\frac{3}{2}$ approximation with time complexity $O(nm\log(mn))$ [Wu, Zhang, and Chen '23]. We present a new practically efficient algorithm with an approximation ratio of $\approx (1.4593 + \varepsilon)$ and a time complexity of $O(nm \log \frac{1}{\varepsilon})$. Our result also applies to the contiguous variant of the problem. In addition to our theoretical results, we implement the presented algorithm and show that the practical performance is significantly better than the theoretical worst-case approximation ratio.

翻译：在可塑作业调度问题中，我们拥有$m$台相同的机器和$n$个作业，每个作业可在可变数量的机器上执行。每个作业的执行时间取决于分配给该作业的机器数量。针对单调可塑作业调度这一具体问题，我们假设作业的处理时间随机器数量的增加而非递增。先前已知的最佳算法包括：(1) 时间复杂度为$Ω(n^{g(1/\varepsilon)})$的多项式时间近似方案，其中$g(\cdot)$是超指数函数[Jansen和Thöle '08; Jansen和Land '18]；(2) 针对$m \geq 8\frac{n}{\varepsilon}$情形的完全多项式时间近似方案[Jansen和Land '18]；以及(3) 近似比为$\frac{3}{2}$、时间复杂度为$O(nm\log(mn))$的算法[Wu, Zhang和Chen '23]。我们提出了一种新的实用高效算法，其近似比约为$(1.4593 + \varepsilon)$，时间复杂度为$O(nm \log \frac{1}{\varepsilon})$。我们的结果也适用于该问题的连续变体。除了理论贡献外，我们还实现了所提出的算法，并证明其实际性能显著优于理论最坏情况下的近似比。