Improved lower bounds for online scheduling to minimize total stretch

In this paper, we show how to construct instances to obtain various lower bounds on the competitive ratio for each m. In the instances, m jobs are given regularly for a sufficiently large time span after a maliciously chosen time. The processing times of the given jobs are taken depending on the remaining processing times of uncompleted jobs at the start time of the “burst” of jobs. We prove that for the instances, the stretch of a job completed before the burst hardly affects the evaluation of a competitive ratio. Further, we provide a job sequence given before the burst for each m. Then, we can improve the previous lower bounds for any deterministic online algorithm for each m using the instances. For example, we obtain lower bounds of 1.228, 1.257 and 21/17≈1.235 for m=1,2 and ∞, respectively. Moreover, we obtain the first non-trivial lower bounds for any randomized online algorithm for all m.