We consider the online scheduling problem of moldable task graphs on multiprocessor systems for minimizing the overall completion time (or makespan). Moldable job scheduling has been widely studied in the literature, in particular when tasks have dependencies (i.e., task graphs) or when tasks are released on-the-fly (i.e., online). However, few studies have focused on both (i.e., online scheduling of moldable task graphs). In this paper, we design a new online scheduling algorithm for this problem and derive constant competitive ratios under several common yet realistic speedup models (i.e., roofline, communication, Amdahl, and a general combination). These results improve the ones we have shown in the preliminary version of this paper. We also prove, for each speedup model, a lower bound on the competitiveness of any online list scheduling algorithm that allocates processors to a task based only on the task's parameters and not on its position in the graph. This lower bound matches exactly the competitive ratio of our algorithm for the roofline, communication and Amdahl's model, and is close to the ratio for the general model. Finally, we provide a lower bound on the competitive ratio of any deterministic online algorithm for the arbitrary speedup model, which is not constant but depends on the number of tasks in the longest path of the graph.

翻译：我们研究了多处理器系统中可塑任务图的在线调度问题，目标是最小化整体完成时间（或制造期）。可塑作业调度在文献中已有广泛研究，特别是涉及任务依赖关系（即任务图）或任务在线释放（即在线场景）的情况。然而，同时考虑这两种因素（即可塑任务图的在线调度）的研究较少。本文针对该问题设计了一种新的在线调度算法，并在几种常见且实际的加速模型（即屋顶线模型、通信模型、阿姆达尔模型及通用组合模型）下推导出常数竞争比。这些结果改进了我们在本文初版中展示的结果。此外，我们针对每种加速模型证明了任何仅基于任务参数而非其在图中位置来分配处理器的在线列表调度算法的竞争比下界。该下界在我们的算法下与屋顶线模型、通信模型和阿姆达尔模型的竞争比完全匹配，且接近通用模型的竞争比。最后，我们针对任意加速模型给出了确定性在线调度算法的竞争比下界，该下界并非常数，而是取决于图中最长路径上的任务数量。