Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors

Scheduling parallel jobs to minimize the makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over tim

Scheduling Parallel Jobs to Minimize Makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a pre-specified, job-dependent number of machines when being processed. Our main result is the following. The makespan of a (non-preemptive) schedule constructed by any listscheduling algorithm is within a factor of 2 of the optimal preemptive makespan. This gives the best known approximation algorithms for both the preemptive and the non-preemptive variant of the problem, improving upon previously known performance guarantees of 3. We also show that no listscheduling algorithm can achieve a better performance guarantee than 2 for the non-preemptive problem, no matter which priority list is chosen. Since listscheduling also works in the online setting in which jobs arrive over time and the length of a job becomes only known when it completes, the main result yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. In this context, no listscheduling algorithm has a constant competitive ratio. We present the first online algorithm for scheduling parallel jobs with a constant competitive ratio. We also prove a new information-theoretic lower bound of 2:25 for the competitive ratio of any deterministic online algorithm for this model

Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors

Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called Hm , where m is a positive integer. Algorithm Hm has a moderate asymptotic worst-case performance ratio in the range [1 3 ::1 18 ] for all m 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1=(r+1)::1+1=r], when task sizes do not exceed 1=r of the total available processors, where r > 1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm Hm is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processor