Parameterized Complexity of Scheduling Chains of Jobs with Delays

H. Bodlaender, M. V. D. Wegen
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引用次数: 6

Abstract

In this paper, we consider the parameterized complexity of the following scheduling problem. We must schedule a number of jobs on $m$ machines, where each job has unit length, and the graph of precedence constraints consists of a set of chains. Each precedence constraint is labelled with an integer that denotes the exact (or minimum) delay between the jobs. We study different cases; delays can be given in unary and in binary, and the case that we have a single machine is discussed separately. We consider the complexity of this problem parameterized by the number of chains, and by the thickness of the instance, which is the maximum number of chains whose intervals between release date and deadline overlap. We show that this scheduling problem with exact delays in unary is $W[t]$-hard for all $t$, when parameterized by the thickness, even when we have a single machine ($m = 1$). When parameterized by the number of chains, this problem is $W[1]$-complete when we have a single or a constant number of machines, and $W[2]$-complete when the number of machines is a variable. The problem with minimum delays, given in unary, parameterized by the number of chains (and as a simple corollary, also when parameterized by the thickness) is $W[1]$-hard for a single or a constant number of machines, and $W[2]$-hard when the number of machines is variable. With a dynamic programming algorithm, one can show membership in XP for exact and minimum delays in unary, for any number of machines, when parameterized by thickness or number of chains. For a single machine, with exact delays in binary, parameterized by the number of chains, membership in XP can be shown with branching and solving a system of difference constraints. For all other cases for delays in binary, membership in XP is open.
时滞作业调度链的参数化复杂度
本文考虑下列调度问题的参数化复杂度。我们必须在$m$机器上调度许多作业,其中每个作业具有单位长度,并且优先约束图由一组链组成。每个优先级约束都标有一个整数,该整数表示作业之间的确切(或最小)延迟。我们研究不同的案例;时滞可以用一元和二进位形式给出,我们单独讨论了单机的情况。我们考虑这个问题的复杂性参数化的链的数量,并通过实例的厚度,这是最大的链的数量,其间隔的发布日期和截止日期重叠。我们表明,这种具有一元精确延迟的调度问题是$W[t]$-当用厚度参数化时,即使我们有一台机器($m = 1$),对于所有$t$来说都是困难的。当用链数参数化时,当我们有单个或恒定数量的机器时,这个问题是$W[1]$-complete,当机器数量是一个变量时,这个问题是$W[2]$-complete。最小延迟的问题,以一元形式给出,由链的数量参数化(作为一个简单的推论,也可以由厚度参数化),对于单个或恒定数量的机器是$W[1]$-hard,当机器数量是可变的时候是$W[2]$-hard。使用动态规划算法,可以在任意数量的机器上,通过厚度或链数参数化,在XP中显示精确和最小一元延迟的隶属关系。对于单个机器,具有精确的二进制延迟,由链数参数化,XP中的成员可以通过分支和求解差分约束系统来表示。对于二进制延迟的所有其他情况,XP中的成员资格是开放的。
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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