Abstract
We introduce a novel adversarial model for scheduling with explorable uncertainty. In this model, the processing time of a job can potentially be reduced (by an a priori unknown amount) by testing the job. Testing a job j takes one unit of time and may reduce its processing time from the given upper limit \(\bar{p}_j\) (which is the time taken to execute the job if it is not tested) to any value between 0 and \(\bar{p}_j\). This setting is motivated e.g., by applications where a code optimizer can be run on a job before executing it. We consider the objective of minimizing the sum of completion times on a single machine. All jobs are available from the start, but the reduction in their processing times as a result of testing is unknown, making this an online problem that is amenable to competitive analysis. The need to balance the time spent on tests and the time spent on job executions adds a novel flavor to the problem. We give the first and nearly tight lower and upper bounds on the competitive ratio for deterministic and randomized algorithms. We also show that minimizing the makespan is a considerably easier problem for which we give optimal deterministic and randomized online algorithms.
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Notes
Files that can be opened with the algebraic solver Mathematica are available at the URL http://cslog.uni-bremen.de/nmegow/public/mathematica-SwT.zip.
We define the problem with rational numbers for the ease of representing them in a computer, but all our results and proofs also hold for real numbers.
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Acknowledgements
We would like to thank Markus Jablonka and Bruno Gaujal for helpful discussions about the algorithm DelayAll, as well as an anonymous referee for pointing us to related work on exploration versus exploitation in the multi-armed bandit framework.
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This research was carried out in the framework of Matheon supported by Einstein Foundation Berlin, the German Science Foundation (DFG) under contract ME 3825/1 and Bayerisch-Französisches Hochschulzentrum (BFHZ). Further support was provided by EPSRC Grant EP/S033483/1 and the ANR Grant ANR-18-CE25-0008. The second author was supported by a study leave granted by University of Leicester during the early stages of the research. A preliminary version of this paper appeared in The 9th Innovations in Theoretical Computer Science Conference (ITCS), January 2018 [16].
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Dürr, C., Erlebach, T., Megow, N. et al. An Adversarial Model for Scheduling with Testing. Algorithmica 82, 3630–3675 (2020). https://doi.org/10.1007/s00453-020-00742-2
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DOI: https://doi.org/10.1007/s00453-020-00742-2