Semi-Online Scheduling on Two Uniform Machines with Known Optimum Part II: Tight Upper Bounds

We consider a semi-online version of the problem of scheduling a sequence of jobs of different lengths on two uniform machines with given speeds 1 and s. Jobs are revealed one by one (the assignment of a job has to be done before the next job is revealed), and the objective is to minimize the makespan. In the considered variant the optimal offline makespan is known in advance. The most studied question for this online-type problem is to determine the optimal competitive ratio, that is, the worst-case ratio of the solution given by an algorithm in comparison to the optimal offline solution. In this paper, we make a further step towards completing the answer to this question by determining the optimal competitive ratio for s between 5+ √ 241 12 ≈ 1.7103 and √ 3 ≈ 1.7321, one of the intervals that were still open. Namely, we present and analyze a compound algorithm achieving the previously known lower bounds. Additionally, we implemented our algorithm and give some experimental results on its average performance. The algorithm applies the method of “safe sets”, with the novel addition that the conditions are not required in a strict way but exceptions are accepted and handled with additional techiques. ∗Department of Mathematics, University of Pannonia, Veszprém, Hungary, [email protected], corresponding author †Helmut Schmidt University / University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany, [email protected] ‡Department of Mathematics, Zhejiang University, Hangzhou, Peoples Republic of China, [email protected] §Department of Computer Science and Systems Technology, University of Pannonia, Veszprém, Hungary, [email protected] ¶Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary ‖Helmut Schmidt University / University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany, [email protected] ∗∗Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland, [email protected]