单台和全同平行机上的总拉伸极小化
Total stretch minimization on single and identical parallel machines
摘要：
我们考虑了经典的问题调度$n$作业的释放日期在单一和相同的并行机。我们通过作业的伸展来度量提供给每个作业的服务质量，该伸展被定义为作业的响应时间与处理时间的比率。我们的目标是非抢先地安排这些工作，以最小化总和拉伸。到目前为止，有很少的结果，特别是对非抢占的情况下，和拉伸最小化。对于抢占式版本，最短剩余处理时间（SRPT）算法是已知的，给$2$ -竞争的和拉伸单机上，而它是$13$ -竞争的相同的并行机。Leonardi和Kellerer分别给出了单机和全同并行机上更一般的\textit{sum (weighted) flow time}问题的强下界。因此，我们研究这个问题，一些额外的假设，并提出了两个新的竞争比现有的算法。证明了最短加工时间（SPT）算法在单机上是$Δ- \frac{1}Δ+1$ -竞争的，在$m$相同的并行机上是$Δ- \frac{1}Δ+ \frac{3}{2} -\frac{1}{2m}$的，其中$Δ$是工件最大加工时间与最小加工时间之比的上界.
英文摘要：
We consider the classical problem of scheduling $n$ jobs with release dates on both single and identical parallel machines. We measure the quality of service provided to each job by its stretch, which is defined as the ratio of its response time to processing time. Our objective is to schedule these jobs non-preemptively so as to minimize sum stretch. So far, there have been very few results for sum stretch minimization especially for the non-preemptive case. For the preemptive version, the Shortest remaining processing time (SRPT) algorithm is known to give $2$-competitive for sum stretch on single machine while its is $13$-competitive on identical parallel machines. Leonardi and Kellerer provided the strong lower bound for the more general problem of \textit{sum (weighted) flow time} in single machine and identical parallel machines, respectively . Therefore, we study this problem with some additional assumptions and present two new competitive ratio for existing algorithms. We show that the Shortest processing time (SPT) algorithm is $Δ- \frac{1}Δ+1$-competitive for non-preemptive sum stretch minimization on single machine and it is $Δ- \frac{1}Δ+ \frac{3}{2} -\frac{1}{2m}$ on $m$ identical parallel machines, where $Δ$ is the upper bound on the ratio between the maximum and the minimum processing time of the jobs.
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