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Approximation Algorithms for Multiprocessor Scheduling with Testing to Minimize the Total Job Completion Time

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Abstract

In offline scheduling models, jobs are given with their exact processing times. In their online counterparts, jobs arrive in sequence together with their processing times and the scheduler makes irrevocable decisions on how to execute each of them upon its arrival. We consider a semi-online variant which has equally rich application background, called scheduling with testing, where the exact processing time of a job is revealed only after a required testing operation is finished, or otherwise the job has to be executed for a given possibly over-estimated length of time. For multiprocessor scheduling with testing to minimize the total job completion time, we present several first approximation algorithms with constant competitive ratios for various settings, including a \(2 \varphi \)-competitive algorithm for the non-preemptive general testing case and a \((0.0382 + 2.7925 (1 - \frac{1}{2\,m}))\)-competitive randomized algorithm, when the number of machines \(m \ge 37\) or otherwise 2.7925-competitive, where \(\varphi = (1 + \sqrt{5}) / 2 < 1.6181\) is the golden ratio and m is the number of machines, a \((3.5 - \frac{3}{2\,m})\)-competitive algorithm allowing job preemption when \(m \ge 3\) or otherwise 3-competitive, and a \((\varphi + \frac{\varphi + 1}{2} (1 - \frac{1}{\,}m))\)-competitive algorithm for the non-preemptive uniform testing case when \(m \ge 5\) or otherwise \((\varphi + 1)\)-competitive. Our results improve three previous best approximation algorithms for the single machine scheduling with testing problems, respectively.

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Notes

  1. In the fully online case, the jobs arrive in sequence.

  2. The authors of [2] in fact examined the more restricted test-preemptive variant in which both testing operations and execution operations are non-preemptive, but for tested job its execution operation does not have to immediately follow the testing operation, nor does it have to be processed on the same machine. Their 2-competitive Threshold algorithm is designed for such a variant, assumed with access to unlimited computational power. Without the access assumption their algorithm is a polynomial-time \((2 + \epsilon )\)-competitive algorithm for any \(\epsilon > 0\).

  3. In the 4-competitive (1, 1)-SORT algorithm, the operations of \(\mathcal {U}\) and \(\mathcal {E}\) are merged in one set and maintained in the non-decreasing order of their processing times.

  4. We integrate such a weighting scheme from the 4-competitive (1, 1)-SORT algorithm, which uses the processing time of an operation of \(\mathcal {U}\cup \mathcal {E}\) to determine their definition of the shortest operation.

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Acknowledgements

The authors are grateful to the reviewers for their many insightful comments and suggestions that greatly improve the presentation. This research is supported by the NSERC Canada and the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan Grant No. 18K11183.

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Authors and Affiliations

  1. Department of Computing Science, University of Alberta, Edmonton, AB, T6G 2E8, Canada

    Mingyang Gong & Kuniteru Hayashi

  2. Division of Information System Design, Tokyo Denki University, Saitama, Japan

    Zhi-Zhong Chen

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  1. Mingyang Gong
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  2. Zhi-Zhong Chen
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Correspondence to Kuniteru Hayashi.

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Gong, M., Chen, ZZ. & Hayashi, K. Approximation Algorithms for Multiprocessor Scheduling with Testing to Minimize the Total Job Completion Time. Algorithmica 86, 1400–1427 (2024). https://doi.org/10.1007/s00453-023-01198-w

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