On-line scheduling with precedence constraints

We consider the on-line problem of scheduling jobs with precedence constraints on m machines. We concentrate in two models, the model of uniformly related machines and the model of restricted assignment. For the related machines model, we show a lower bound of Ω(√m) for the competitive ratio of deterministic and randomized on-line algorithms, with or without preemptions even for known running times. This matches the deterministic upper bound of O(√m) given by Jaffe. The lower bound should be contrasted with the known bounds for jobs without precedence constraints. Specifically, without precedence constraints, if we allow preemptions then the competitive ratio becomes Θ(log m), and if the running times of the jobs are known then there are O(1) competitive (preemptive and non-preemptive) algorithms. We also consider the restricted assignment model. For the model with consistent precedence constraints, we give a (randomized) lower bound of Ω(log m) with or without preemptions. We show that a (deterministic, non-preemptive) greedy algorithm is optimal up to a constant factor for this model i.e. O(log m) competitive. However, for general precedence constraints, we show a lower bound of m which is easily matched by a greedy algorithm.