Optimal on-line flow time with resource augmentation

AbstractWe 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 ℓ machines. We design an algorithm of competitive ratio 1+2min(Δ1/ℓ,n1/ℓ), where Δ 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 ℓ. 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 Δ 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 ℓm 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

Trade-offs between speed and processor in hard-deadline scheduling

This paper revisits the problem of on-line scheduling of sequential jobs with hard deadlines in a preemptive, multiprocessor setting. An on-line scheduling algorithm is said to be optimal if it can schedule any set of jobs to meet their deadlines whenever it is feasible in the off-line sense. It is known that the earliest-deadline-first strategy (EDF) is optimal in a one-processor setting, and there is no optimal on-line algorithm in an m-processor setting where m≥2. Recent work however reveals that if the on-line algorithm is given faster processors, EDF is actually optimal for all m (e.g., when m = 2, it suffices to use processors 1.5 times as fast). This paper initiates the study of the trade-off between increasing the speed and using more processors in deriving optimal on-line scheduling algorithms. Several upper bound and lower bound results are presented. For example, the speed requirement of EDF can be reduced to 2-1+p/m+p when it is given p≥0 extra processors. The main result is a new on-line algorithm which demands less speedy processors so as to attain optimality (e.g., when m = 2, the speed requirement is 1 1/3) and admits a better speed-processor trade-off than EDF (e.g., when m = 2 and p = 1, the speed requirement is 1.2). In general, no optimal algorithm exists when the speed factor is less than 1/(2√2+p/m-2).published_or_final_versio

New results on 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 $l$ machines. We design an algorithm of competitive ratio O(min(Delta^{1/l,n^{1/l)), where $Delta$ 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 $l$. 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 $Delta$ is 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 $lm$ 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

Optimal time-critical scheduling via resource augmentation

We consider two fundamental problems in dynamic scheduling: scheduling to meet deadlines in a preemptive multiprocessor setting, and scheduling to provide good response time in a number of scheduling environments. When viewed from the perspective of traditional worst-case analysis, no good on-line algorithms exist for these problems, and for some variants no good off-line algorithms exist unless {Rho} = {Nu}{Rho}. We study these problems using a relaxed notion of competitive analysis, introduced by Kalyanasundaram and Pruhs, in which the on-line algorithm is allowed more resources than the optimal off-line algorithm to which it is compared. Using this approach, we establish that several well-known on-line algorithms, that have poor performance from an absolute worst-case perspective, are optimal for the problems in question when allowed moderately more resources. For the optimization of average flow time, these are the first results of any sort, for any {Nu}{Rho}-hard version of the problem, that indicate that it might be possible to design good approximation algorithms

Approximation Techniques for Average Completion Time Scheduling

We consider the problem of nonpreemptive scheduling to minimize average ( weighted) completion time, allowing for release dates, parallel machines, and precedence constraints. Recent work has led to constant-factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs. We introduce several new techniques which generalize this basic paradigm. We use these ideas to obtain improved approximation algorithms for one-machine scheduling to minimize average completion time with release dates. In the process, we obtain an optimal randomized on-line algorithm for the same problem that beats a lower bound for deterministic on-line algorithms. We consider extensions to the case of parallel machine scheduling, and for this we introduce two new ideas: first, we show that a preemptive one-machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times; second, we show that a nongreedy “rounding” of the relaxation yields better approximations than a greedy one. We also prove a general theore mrelating the value of one- machine relaxations to that of the schedules obtained for the original m-machine problems. This theorem applies even when there are precedence constraints on the jobs. We apply this result to obtain improved approximation ratios for precedence graphs such as in-trees, out-trees, and series-parallel graphs