# On the parameterized tractability of the just-in-time flow-shop scheduling problem

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## Abstract

We study the parameterized complexity of a set of flow-shop scheduling problems in which the objective is to maximize the weighted number of just-in-time jobs. Our analysis focuses on the case where the number of due dates is considerably smaller than the number of jobs and thus can be considered as a parameter. We prove that the two-machine problem is W[1]-hard with respect to this parameter, even if all processing times on the second machine are of unit length, while the problem is in XP when parameterized by the number of machines combined with the number of different due dates. We then move on to study the tractability of the problem when combining the number of different due dates with either the number of different weights or the number of different processing times on the first machine. We prove that in both cases the problem is fixed-parameter tractable for the two-machine case and is W[1]-hard for three or more machines.

## Keywords

Scheduling Flow-shop Just-in-time Parameterized complexity Fixed-parameter tractability## Notes

### Acknowledgements

This research was partially supported by Grant No. 2016049 from the United States-Israel Binational Science Foundation (BSF). The first author’s work was supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant 631163.11, and by the Israel Science Foundation (Grant 551145/14). We would also like to thank the reviewers for their insightful comments on this paper, comments which improved the paper dramatically.

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