Two semi-online scheduling problems on two uniform machines

This paper considers two semi-online scheduling problems, one with known optimal value and the other with known total sum, on two uniform machines with a machine speed ratio of s ≥ 1. For the first problem, we provide an optimal algorithm for s ∈ [frac(1 + sqrt(3), 2), frac(1 + sqrt(21), 4)], and improved algorithms or/and lower bounds for s ∈ [frac(1 + sqrt(21), 4), sqrt(3)], over which the optimal algorithm is unknown. As a result, the largest gap between the competitive ratio and the lower bound decreases to 0.02192. For the second problem, we also present algorithms and lower bounds for s ≥ 1. The largest gap between the competitive ratio and the lower bound is 0.01762, and the length of the interval over which the optimal algorithm is unknown is 0.47382. Our algorithms and lower bounds for these two problems provide insights into their differences, which are unusual from the viewpoint of the known results on these two semi-online scheduling problems in the literature.