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

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 $$\frac{5 + \sqrt{241}}{12} \approx 1.7103$$ 5 + 241 12 ≈ 1.7103 and $$\sqrt{3} \approx 1.7321$$ 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