Scheduling with testing falls under the umbrella of the research on optimization with explorable uncertainty. In this model, each job has an upper limit on its processing time that can be decreased to a lower limit (possibly unknown) by some preliminary action (testing). Recently, D{\"{u}}rr et al. \cite{DBLP:journals/algorithmica/DurrEMM20} has studied a setting where testing a job takes a unit time, and the goal is to minimize total completion time or makespan on a single machine. In this paper, we extend their problem to the budget setting in which each test consumes a job-specific cost, and we require that the total testing cost cannot exceed a given budget. We consider the offline variant (the lower processing time is known) and the oblivious variant (the lower processing time is unknown) and aim to minimize the total completion time or makespan on a single machine. For the total completion time objective, we show NP-hardness and derive a PTAS for the offline variant based on a novel LP rounding scheme. We give a $(4+\epsilon)$-competitive algorithm for the oblivious variant based on a framework inspired by the worst-case lower-bound instance. For the makespan objective, we give an FPTAS for the offline variant and a $(2+\epsilon)$-competitive algorithm for the oblivious variant. Our algorithms for the oblivious variants under both objectives run in time $O(poly(n/\epsilon))$. Lastly, we show that our results are essentially optimal by providing matching lower bounds.

翻译：具有测试机制的调度问题属于可探索不确定性优化研究的范畴。在该模型中，每个作业的处理时间存在一个上限，可通过初步操作（测试）降低至一个（可能未知的）下限。最近，Dürr等人在文献\cite{DBLP:journals/algorithmica/DurrEMM20}中研究了测试一个作业需单位时间、且目标为最小化单机总完工时间或最大完工时间的设置。本文将其问题扩展至预算场景：每次测试消耗作业特定的成本，且总测试成本不得超过给定预算。我们考虑离线变体（下限处理时间已知）与不知情变体（下限处理时间未知），旨在最小化单机上的总完工时间或最大完工时间。针对总完工时间目标，我们证明了问题的NP困难性，并基于新颖的线性规划舍入方案为离线变体提出了多项式时间近似方案。我们为不知情变体设计了一个基于最坏情况下界实例启发式框架的$(4+\epsilon)$-竞争比算法。针对最大完工时间目标，我们为离线变体提出了完全多项式时间近似方案，并为不知情变体给出了$(2+\epsilon)$-竞争比算法。针对两个目标所提出的不知情变体算法均在$O(poly(n/\epsilon))$时间内运行。最后，我们通过匹配下界证明所提结果在本质上达到最优。