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Tight upper bounds for semi-online scheduling on two uniform machines with known optimum |
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| Authors: | Dósa György Fügenschuh Armin Tan Zhiyi Tuza Zsolt Węsek Krzysztof |
| Institution: | 1.Department of Mathematics, University of Pannonia, Veszprém, Hungary ;2.Helmut Schmidt University/University of the Federal Armed Forces Hamburg, Holstenhofweg 85, 22043, Hamburg, Germany ;3.Department of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China ;4.Department of Computer Science and Systems Technology, University of Pannonia, Veszprém, Hungary ;5.Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary ;6.Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662, Warsaw, Poland ; |
| Abstract: | 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 \(\frac{5 + \sqrt{241}}{12} \approx 1.7103\) and \(\sqrt{3} \approx 1.7321\), one of the intervals that were still open. Namely, we present and analyze a compound algorithm achieving the previously known lower bounds. |
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