On preemptive scheduling on unrelated machines using linear programming

AIMS Mathematics

article

Nodari Vakhania¹ 

[1] Centro de Investigación en Ciencias, Universidad Autónoma del Estado de Morelos

关键词: scheduling;    unrelated machines;    release time;    linear programming;    time complexity;   

DOI  :  10.3934/math.2023356

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

We consider a basic preemptive scheduling problem where $ n $ non-simultaneously released jobs are to be processed by $ m $ unrelated parallel machines so as to minimize maximum job completion time. An optimal LP-solution has been used to construct an optimal preemptive schedule for simultaneously released jobs in time $ O(n^3) $. We propose fast $ O(m) $ time algorithm that finds an optimal schedule in case the above LP-solution possesses "small enough" number of non-zero elements. We propose another linear program for non-simultaneously released jobs and show how an optimal schedule can be constructed also in time $ O(m) $ from the optimal solution to that linear program. Based on another stronger linear program formulation, we extend the earlier known schedule construction procedure for non-simultaneously released jobs. The procedure is important, in particular, because there may exist no optimal schedule that agrees with an optimal LP-solution. An optimal LP-solution imposes a number of preemptions, and additional preemptions may occur during the schedule construction process, a job might be forced to be split on the same machine. We show that if no job split is allowed, even a restricted version of the problem on three unrelated machines is NP-hard. As a result, we obtain that, given an optimal LP-solution, it is NP-hard to find an optimal schedule that agrees with that LP-solution. As another side result, we obtain that it is NP-hard to find an optimal schedule with at most $ m-1 $ preemptions.

【 授权许可】

CC BY