We study online scheduling to minimize total completion time with explorable uncertainty on single and multiple machines. Each job comes with an upper limit of its processing time, which could be potentially reduced by testing the job, which also takes time. The objective is to schedule all jobs with minimum total completion time. The challenge lies in deciding which jobs to test, the order of testing/processing jobs, and in multiple machine case which machine a job is allocated to. In multiple machine case, testing and processing of a job are allowed to be scheduled on different machines. Different settings have been studied before. In this work, we first consider the variable testing times setting. We enhance the analysis framework in Albers and Eckl (2020) and improve the analysis of the competitive ratio of their deterministic single machine algorithm from $4$ to $1+\sqrt{2} \approx 2.4143$. Using the new analysis framework, we propose a new deterministic algorithm that further improves the competitive ratio to $2.316513$. The new framework also enables us to develop a randomized algorithm improving the expected competitive ratio from $3.3794$ to $2.152271$. We further show that with $m$~machines, by extending the framework of Gong et al. (2024), there exists a deterministic $2.77629-(0.45977/m)$-competitive algorithm and a randomized $2.51098-(0.3587/m)$-competitive algorithm. The performance of the algorithms on multiple machines when $m = 1$ matches the current best algorithms on a single machine for variable testing times shown in this paper.

翻译：我们研究在线调度以最小化具有可探测不确定性的单机和多机总完成时间。每个作业附带其处理时间的上限，该时间可通过测试作业（同样消耗时间）潜在缩短。目标是调度所有作业并最小化总完成时间。挑战在于决定哪些作业需要测试、测试/处理作业的顺序，以及在多机情况下作业分配到哪台机器。在多机情况下，允许将同一作业的测试和处理安排在不同机器上。已有研究探讨了不同设置。本文首先考虑可变测试时间设置。我们增强了Albers和Eckl（2020）的分析框架，将其确定性单机算法竞争比的分析从$4$改进至$1+\sqrt{2} \approx 2.4143$。基于新分析框架，我们提出一种新的确定性算法，进一步将竞争比提升至$2.316513$。新框架还使我们能够开发随机算法，将期望竞争比从$3.3794$改进至$2.152271$。进一步证明，对于$m$台机器，通过扩展Gong等人（2024）的框架，存在确定性$2.77629-(0.45977/m)$-竞争算法和随机$2.51098-(0.3587/m)$-竞争算法。当$m=1$时，多机算法性能与本文展示的针对可变测试时间的当前最优单机算法相匹配。