Multiprocessor scheduling, also called scheduling on parallel identical machines to minimize the makespan, is a classic optimization problem which has been extensively studied. Scheduling with testing is an online variant, where the processing time of a job is revealed by an extra test operation, otherwise the job has to be executed for a given upper bound on the processing time. Albers and Eckl recently studied the multiprocessor scheduling with testing; among others, for the non-preemptive setting they presented an approximation algorithm with competitive ratio approaching 3.1016 when the number of machines tends to infinity and an improved approximation algorithm with competitive ratio approaching 3 when all test operations take one unit of time each. We propose to first sort the jobs into non-increasing order of the minimum value between the upper bound and the testing time, then partition the jobs into three groups and process them group by group according to the sorted job order. We show that our algorithm achieves better competitive ratios, which approach 2.9513 when the number of machines tends to infinity in the general case; when all test operations each takes one time unit, our algorithm achieves even better competitive ratios approaching 2.8081.

多处理器调度，也称为并行相同机器上的调度以最小化制造时间，是一个经典的优化问题，已被广泛研究。带测试的调度是一种在线变体，其中作业的处理时间由额外的测试操作显示，否则必须在给定的处理时间上限内执行作业。Albers 和 Eckl 最近通过测试研究了多处理器调度；其中，对于非抢占式设置，他们提出了一种近似算法，当机器数量趋于无穷时，竞争比接近 3.1016；当所有测试操作每个都花费一个单位时间时，竞争比接近 3 的改进近似算法。我们建议首先将作业排序为上限和测试时间之间最小值的非递增顺序，然后将作业分成三组，按照排序后的作业顺序逐组处理。我们表明我们的算法实现了更好的竞争比率，当机器数量在一般情况下趋于无穷大时，接近 2.9513；当所有测试操作都需要一个时间单位时，我们的算法实现了更好的竞争比率，接近 2.8081。