LP-Based Online Scheduling: From Single to Parallel Machines

We study classic machine sequencing problems in an online setting. Specifically, we look at deterministic and randomized algorithms for the problem of scheduling jobs with release dates on identical parallel machines, to minimize the sum of weighted completion times: Both preemptive and non-preemptive versions are analyzed. Using linear programming techniques, borrowed from the single machine case, we are able to design a 2.62-competitive deterministic algorithm for the nonpreemptive version of the problem, improving upon the 3.28-competitive algorithm of Megow and Schulz [13]. Additionally, we obtain randomized algorithms for both versions of the problem with competitive ratio strictly smaller than 2 for any number of machines (but approaching two as the number of machines grows). Our algorithms naturally extend several approaches for single and parallel machine scheduling. Moreover, in contrast to previous algorithms achieving similar bounds, obtained by Schulz and Skutella [15], ours is a list-scheduling algorithm and has one less randomization step.