# Tight lower bounds for semi-online scheduling on two uniform machines with known optimum

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## Abstract

This problem is about scheduling a number of jobs on two uniform machines with given speeds 1 and \(s\ge 1\), so that the overall finishing time, i.e., the makespan, is earliest possible. We consider a semi-online variant (introduced for equal speeds) by Azar and Regev, where the jobs are arriving one after the other, while the scheduling algorithm knows the optimum value of the corresponding offline problem. One can ask how close any possible algorithm could get to the optimum value, that is, to give a lower bound on the competitive ratio: the supremum over ratios between the value of the solution given by the algorithm and the optimal offline solution. We contribute to this question by constructing tight lower bounds for all values of *s* in the intervals \([\frac{1+\sqrt{21}}{4},\frac{3+\sqrt{73}}{8}]\approx [1.3956,1.443]\) and \([\frac{5}{3},\frac{4+\sqrt{133}}{9}]\approx [\frac{5}{3},1.7258]\), except a very narrow interval.

## Keywords

Semi-online scheduling Makespan minimization Machine scheduling Lower bounds## Notes

### Acknowledgements

Gyorgy Dosa acknowledges the financial support of Szechenyi 2020 under the EFOP-3.6.1-16-2016-00015. György Dósa’s and Zsolt Tuza’s work was jointly funded by the National Research, Development and Innovation Office – NKFIH under the grant SNN 116095. Zhiyi Tan’s work was supported by the National Natural Science Foundation of China (11671356, 11271324, 11471286). Krzysztof Węsek’s work was partially supported by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme, realized by Center for Advanced Studies. Furthermore, Węsek’s work was conducted as a guest researcher at the Helmut Schmidt University. Armin Fügenschuh’s work was mostly carried out while being affiliated with the Helmut Schmidt University. Fügenschuh acknowledges the support by the German Research Association (DFG), grant number FU 860/1-1. Last but not least, we are grateful to the two anonymous referees for their various helpful comments on our manuscript.

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