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 Omega(rootm) for deterministic and randomized on-line algorithms, with or without preemptions even for jobs of known durations. This matches the deterministic upper bound of O(rootm) given by Jaffe for task systems. 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 O(log m), and if the durations 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 Omega (log m) with or without preemptions. We show that the (deterministic) greedy algorithm (no preemptions used), is optimal for this model i.e. O(log m) competitive. However, far general precedence constraints, we show a lower bound of rn which is easily matched by a greedy algorithm.