优先序约束的排序问题：基于最大匹配的近似算法

doi:10.15960/j.cnki.issn.1007-6093.2022.03.005

摘要/Abstract

摘要：

本文研究具有加工次序约束的单位工件开放作业和流水作业排序问题，目标函数为极小化工件最大完工时间。工件之间的加工次序约束关系可以用一个被称为优先图的有向无圈图来刻画。当机器数作为输入时，两类问题在一般优先图上都是强NP-困难的，而在入树的优先图上都是可解的。我们利用工件之间的许可对数获得了问题的新下界，并基于许可工件之间的最大匹配设计近似算法，其中匹配的许可工件对均能同时在不同机器上加工。对于一般优先图的开放作业问题和脊柱型优先图的流水作业问题，我们在理论上证明了算法的近似比为$2-\frac 2m$，其中$m$是机器数目。

关键词: 工件序, 开放作业, 流水作业, 最大匹配, 近似算法

Abstract:

We investigate the problem to schedule a set of precedence constrained jobs of unit size on an open-shop or on a flow-shop to minimize the makespan. The precedence constraints among the jobs are presented as a directed acyclic graph called the precedence graph. When the number of machines in the shop is part of the input, both problems are strongly NP-hard on general precedence graphs, but were proven polynomial-time solvable for some special precedence graphs such as intrees. In this paper, we characterize improved lower bounds on the minimum makespan in terms of the number of agreeable pairings among certain jobs and propose approximation algorithms based on a maximum matching among these jobs, so that every agreeable pair of jobs can be processed on different machines simultaneously. For open-shop with an arbitrary precedence graph and for flow-shop with a spine precedence graph, both achieved approximation ratios are $2 - \frac 2m$, where $m$ is the number of machines in the shop.

Key words: job precedence, open-shop, flow-shop, maximum matching, approximation algorithm

中图分类号:

O221.7

张安, 陈永, 陈光亭, 陈占文, 舒巧君, 林国辉. 优先序约束的排序问题：基于最大匹配的近似算法[J]. 运筹学学报, 2022, 26(3): 57-74.

An ZHANG, Yong CHEN, Guangting CHEN, Zhanwen CHEN, Qiaojun SHU, Guohui LIN. Maximum matching based approximation algorithms for precedence constrained scheduling problems[J]. Operations Research Transactions, 2022, 26(3): 57-74.

图/表 6

Table 1

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Problem		Complexity	Approximation
$ P \mid prec \mid C_{\max} $		NP-hard^[2]	$ 2 - \frac 1m $^[2]
$ P \mid prec, p_j = 1 \mid C_{\max} $		NP-hard to $ 2 $-approx^[8]	$ 2 - \frac 2m $^{[4, 5]}
$ P \mid prec \mid C_{\max} $, $ m {\geqslant} 4 $		NP-hard	$ 2 - \frac 7{3m+1} $^[6]
	$ m {\geqslant} 3 $	NP-hard open^[9]
$ Pm \mid prec, p_j = 1 \mid C_{\max} $	$ m {\geqslant} 4 $	PTAS open^[10]
	$ m {\geqslant} 4 $	Not APX-hard^[11]
$ O \mid prec, p_{ij} = 1\mid C_{\max} $		Strongly NP-hard^{[13, 14]}	$ 2 - \frac 2m $ (^[19] and this paper)
$ F \mid prec, p_{ij} = 1\mid C_{\max} $		Strongly NP-hard^{[13, 14]}
$ F \mid spine, p_{ij} = 1\mid C_{\max} $		NP-hard open	$ 2 - \frac 2m $ (this paper)
$ O/F \mid intree/outtree, p_{ij} = 1\mid C_{\max} $		P^{[16, 17]}
$ O/F \mid intree, p_{ij} = 1\mid L_{\max} $		P^{[15, 16]}
$ O/F \mid outtree, p_{ij} = 1\mid L_{\max} $		Strongly NP-hard^{[14, 18]}
$ Om/Fm \mid prec, p_{ij} = 1 \mid C_{\max} $, $ m {\geqslant} 3 $		NP-hard open
$ O2/F2 \mid prec, p_{ij} = 1 \mid L_{\max} $		P^{[15, 16]}

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