The competitiveness of on-line assignments
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Scheduling Parallel Machines On-line
SIAM Journal on Computing
Online computation and competitive analysis
Online computation and competitive analysis
Algorithms for Scheduling Tasks on Unrelated Processors
Journal of the ACM (JACM)
Lower Bounds for On-line Scheduling with Precedence Constraints on Identical Machines
APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
Developments from a June 1996 seminar on Online algorithms: the state of the art
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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 deterministic and randomized on-line algorithms, with or without preemptions even for jobs of known durations. This matches the deterministic upper bound of O(√m) 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 Θ(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 Ω(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, for general precedence constraints, we show a lower bound of m which is easily matched by a greedy algorithm.