单工件单资源的并行机排序问题
Parallel Machine Scheduling with a Single Resource per Job
摘要：
本文研究了使总完工时间最小的并行机上的工件排序问题，每个工件只占用一个资源。首先，我们推导出问题的基本性质，并证明了该问题是多项式可解的，如果$p_j = 1$。然后，我们看看一个变种的最短处理时间的规则作为一个近似算法的问题，并表明它至少给出了一个$(2-\frac{1}{m})$近似。随后，我们表明，虽然问题的复杂性仍然是开放的，三个相关的问题是$\mathcal{NP}$ -硬。在第一个问题中，每个资源也有一个机器子集，它可以在这些机器上使用。在第二个问题中，一旦一个资源已经在一台机器上使用，它就不能在任何其他机器上使用，因此使用相同资源的所有作业都需要在同一台机器上调度。在第三个问题中，每个作业都需要两个资源，而不是一个。
英文摘要：
We study the problem of scheduling jobs on parallel machines minimizing the total completion time, with each job using exactly one resource. First, we derive fundamental properties of the problem and show that the problem is polynomially solvable if $p_j = 1$. Then we look at a variant of the shortest processing time rule as an approximation algorithm for the problem and show that it gives at least a $(2-\frac{1}{m})$-approximation. Subsequently, we show that, although the complexity of the problem remains open, three related problems are $\mathcal{NP}$-hard. In the first problem, every resource also has a subset of machines on which it can be used. In the second problem, once a resource has been used on a machine it cannot be used on any other machine, hence all jobs using the same resource need to be scheduled on the same machine. In the third problem, every job needs exactly two resources instead of just one.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
