Optimal semi-online preemptive algorithms for machine covering on two uniform machines

In this paper, we consider the semi-online preemptive scheduling problem with decreasing job sizes on two uniform machines. Our goal is to maximize the continuous period of time (starting from time zero) when both machines are busy, which is equivalent to maximizing the minimum machine completion time if idle time is not introduced before all the jobs are completed. We design optimal deterministic semi-online algorithms for every machine speed ratio s∈[1,∞), and show that idle time is required during the assignment procedure of algorithms for any s>6/2. The competitive ratios of the algorithms match the randomized lower bound for every 1⩽s⩽3. The problem of whether randomization still does not help for the discussed preemptive scheduling problem remains open.