SRPT Is 1.86-competitive for Completion Time Scheduling

We consider the classical problem of scheduling preemptible jobs, that ar-rive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj,pmtn | j cj. A pop-ular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [12, 13]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ra-tio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [10]

An Improved Analysis of SRPT Scheduling Algorithm on the Basis of Functional Optimization *

Abstract The competitive performance of the SRPT scheduling algorithm has been open for a long time except for being 2-competitive, where the objective is to minimize the total completion time. Chung et al. proved that the SRPT algorithm is 1.857-competitive. In this paper we improve their analysis and show a 1.792-competitiveness. We clearly mention that our result is not the best so far, since Sitters recently proved the algorithm is 1.250-competitive. Nevertheless, it is still well worth reporting our analytical method; our analysis is based on the modern functional optimization, which can scarcely be found in the literature on the analysis of algorithms. Our aim is to illustrate the potentiality of functional optimization with a concrete application

SRPT is 1.86-Competitive for Completion Time Scheduling

We consider the classical problem of scheduling preemptible jobs, that arrive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj,pmtn | P j cj. A popular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [13, 14]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ratio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [11]