Optimal on-line flow time with resource augmentation

We study the problem of scheduling n   jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ℓℓ machines. We design an algorithm of competitive ratio 1+2min(Δ1/ℓ,n1/ℓ), where ΔΔ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ℓℓ. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only ΔΔ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m   machines and the on-line algorithm has ℓmℓm machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.