We study online scheduling with obligatory testing on $m$ identical machines with the objective of minimizing the sum of completion times. In this model, every job must undergo a test before its actual processing time is revealed. Consequently, the central algorithmic challenge is no longer whether to acquire information, but how to optimally balance machine capacity between revealing unknown jobs and processing currently known ones. While this tradeoff becomes structurally richer in the multiple-machine setting, the only prior explicit deterministic lower bound for this objective was $\sqrt{2}$, established strictly for a single machine in 2024 by Dogeas et al. [ESA 2024: 48:1-48:14]. Our core conceptual contribution is demonstrating that completion-threshold quantities, denoted $T_X$, serve as the fundamental analytical metric for this setting. Because every completed job must first pass through the testing phase, delayed revelation inherently forces delayed completion. By bounding these $T_X$ thresholds, we systematically derive strong lower bounds on the total completion time. Utilizing this framework, we establish the first substantial deterministic lower bounds for multiple machines, including a three-type bound of $1.4811$ and a multi-type dyadic construction that asymptotically approaches $3/2$. Finally, we complement these theoretical limits with a deterministic $2$-competitive list-scheduling algorithm for arbitrary test times.

翻译：我们研究在 $m$ 台同构机器上带有强制测试的在线调度问题,目标是最小化总完工时间。在此模型中,每个作业的实际处理时间必须在测试之后才能揭示。因此,核心算法挑战不再是是否获取信息,而是如何最优地在揭示未知作业与处理当前已知作业之间平衡机器容量。虽然这种权衡在多机环境下在结构上更为丰富,但此前对该目标唯一的显式确定性下界是 $\sqrt{2}$,该下界严格针对单机情形,由 Dogeas 等人在 2024 年建立 [ESA 2024: 48:1-48:14]。我们的核心概念贡献在于证明:完成阈值量(记为 $T_X$)是此环境下的基本分析度量。由于每个完成的作业必须首先经过测试阶段,延迟揭示必然导致延迟完成。通过界定这些 $T_X$ 阈值,我们系统地推导出总完工时间的强下界。利用该框架,我们建立了多机情形下首个重要的确定性下界,包括一个值为 $1.4811$ 的三类型下界以及一个渐近逼近 $3/2$ 的多类型二进构造。最后,作为这些理论极限的补充,我们提出一个适用于任意测试时间的确定性 $2$-竞争列表调度算法。