Abstract
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.
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In fact, the authors of Albers and Eckl (2021) also examined the more restricted test-preemptive variant in which a job can be tested non-preemptively on one machine to obtain its accurate processing time, and then processed non-preemptively on any other machine starting at any later time. Their 2-competitive algorithm with access to unlimited computational power, or polynomial-time \((2 + \epsilon )\)-competitive algorithm for any \(\epsilon > 0\), is designed for such a variant.
We remark that the function \(T^g_m\) as defined in Eq. (1) is explicitly designed to satisfy this equality.
We remark that the function \(T^g_2\) as defined in Eq. (1) is explicitly designed to satisfy this equality.
We remark that the function \(T^u_m\) as defined in Eq. (2) is explicitly designed to satisfy this equality.
We remark that the function \(T^u_2\) as defined in Eq. (2) is explicitly designed to satisfy this equality.
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Acknowledgements
The authors are grateful to the reviewers for their many helpful comments and suggestions. RG and GL are supported by the NSERC Canada. EM is supported by the KAKENHI Grants JP21K11755 and JP17K00016 and the JST CREST JPMJR1402.
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Gong, M., Goebel, R., Lin, G. et al. Improved approximation algorithms for non-preemptive multiprocessor scheduling with testing. J Comb Optim 44, 877–893 (2022). https://doi.org/10.1007/s10878-022-00865-y
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DOI: https://doi.org/10.1007/s10878-022-00865-y