Motivated by modern parallel computing applications, we consider the problem of scheduling parallel-task jobs with heterogeneous resource requirements in a cluster of machines. Each job consists of a set of tasks that can be processed in parallel, however, the job is considered completed only when all its tasks finish their processing, which we refer to as "synchronization" constraint. Further, assignment of tasks to machines is subject to "placement" constraints, i.e., each task can be processed only on a subset of machines, and processing times can also be machine dependent. Once a task is scheduled on a machine, it requires a certain amount of resource from that machine for the duration of its processing. A machine can process ("pack") multiple tasks at the same time, however the cumulative resource requirement of the tasks should not exceed the machine's capacity. Our objective is to minimize the weighted average of the jobs' completion times. The problem, subject to synchronization, packing and placement constraints, is NP-hard, and prior theoretical results only concern much simpler models. For the case that migration of tasks among the placement-feasible machines is allowed, we propose a preemptive algorithm with an approximation ratio of $(6+\epsilon)$. In the special case that only one machine can process each task, we design an algorithm with improved approximation ratio of $4$. Finally, in the case that migrations (and preemptions) are not allowed, we design an algorithm with an approximation ratio of $24$. Our algorithms use a combination of linear program relaxation and greedy packing techniques. We present extensive simulation results, using a real traffic trace, that demonstrate that our algorithms yield significant gains over the prior approaches.

中文翻译：

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调度受包装和放置约束的并行任务作业

受现代并行计算应用程序的启发，我们考虑在机器集群中调度具有异构资源需求的并行任务作业的问题。每个作业由一组可以并行处理的任务组成，但是，只有当其所有任务完成处理时才认为该作业已完成，我们将其称为“同步”约束。此外，将任务分配给机器受“放置”约束，即每个任务只能在机器的子集上处理，并且处理时间也可以取决于机器。一旦任务在一台机器上被调度，它在处理期间需要该机器的一定数量的资源。一台机器可以同时处理（“打包”）多个任务，但是，任务的累积资源需求不应超过机器的容量。我们的目标是最小化作业完成时间的加权平均值。该问题受同步、打包和放置约束的约束，是 NP-hard 问题，并且先前的理论结果仅涉及更简单的模型。对于允许在可放置机器之间迁移任务的情况，我们提出了一种近似比为 $(6+\epsilon)$ 的抢占式算法。在只有一台机器可以处理每个任务的特殊情况下，我们设计了一个改进的近似比为 $4$ 的算法。最后，在不允许迁移（和抢占）的情况下，我们设计了一个近似比率为 $24$ 的算法。我们的算法结合使用线性程序松弛和贪婪打包技术。我们使用真实的交通轨迹展示了广泛的模拟结果，证明我们的算法比之前的方法产生了显着的收益。