Computational complexity and algorithms for two scheduling problems under linear constraints

IF 0.9 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Kameng Nip, Peng Xie
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Abstract

This paper considers two different types of scheduling problems under linear constraints. The first is the single-machine scheduling problem with minimizing total completion time, while the second is the no-wait two-machine flow shop scheduling problem with minimizing makespan. For these two problems, a set of jobs is required to be scheduled to one or two machines. In contrast to the classic scheduling problems, the processing times of jobs are not fixed constants and are not predetermined. The decision-maker only knows that they should satisfy a system of given linear constraints. For both problems, the goal is to determine the processing time for each job and find the schedule that minimizes a particular criterion, namely, the total completion time or the makespan. First, we study the computational complexity and show that both the problems under linear constraints are NP-hard. These hardness results significantly differ from their traditional scheduling counterparts, as both of those are solvable in polynomial time. Then we propose polynomial time exact or approximation algorithms for various special cases. By utilizing the existing scheduling algorithms and the properties of linear programming, we demonstrate that both problems are polynomially solvable when the total number of linear constraints is a fixed constant.

Abstract Image

线性约束下两个调度问题的计算复杂性和算法
本文考虑了线性约束下两种不同类型的调度问题。第一种是总完成时间最小化的单机调度问题,第二种是总时间最小化的无等待双机流程车间调度问题。在这两个问题中,要求将一组作业安排到一台或两台机器上。与传统的调度问题不同,作业的处理时间不是固定不变的,也不是预先确定的。决策者只知道他们应该满足给定的线性约束系统。对于这两个问题,目标都是确定每个作业的处理时间,并找到最小化特定标准(即总完成时间或工期)的计划。首先,我们研究了计算复杂性,结果表明线性约束下的这两个问题都是 NP 难问题。这些难度结果与传统的调度问题有很大不同,因为这两个问题都可以在多项式时间内求解。然后,我们针对各种特殊情况提出了多项式时间精确或近似算法。利用现有的调度算法和线性规划的特性,我们证明了当线性约束的总数是一个固定常数时,这两个问题都是多项式可解的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
期刊介绍: The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.
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