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Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines
arXiv - CS - Computational Complexity Pub Date : 2020-09-24 , DOI: arxiv-2009.11840
Martin Kouteck\'y and Johannes Zink

The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number $k$ of job types, but possibly the number of jobs $n$ is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines ($Q|HM|C_{\max}$) is NP-hard already with $6$ job types, and that the related Cutting Stock problem is NP-hard already with $8$ item types. For the more general unrelated machines model ($R|HM|C_{\max}$), we show that if either the largest job size $p_{\max}$, or the number of jobs $n$ are polynomially bounded in the instance size $|I|$, there are algorithms with complexity $|I|^{\textrm{poly}(k)}$. Our main result is that this is unlikely to be improved, because $Q||C_{\max}$ is W[1]-hard parameterized by $k$ already when $n$, $p_{\max}$, and the numbers describing the speeds are polynomial in $|I|$; the same holds for $R|HM|C_{\max}$ (without speeds) when the job sizes matrix has rank $2$. Our positive and negative results also extend to the objectives $\ell_2$-norm minimization of the load vector and, partially, sum of weighted completion times $\sum w_j C_j$. Along the way, we answer affirmatively the question whether makespan minimization on identical machines ($P||C_{\max}$) is fixed-parameter tractable parameterized by $k$, extending our understanding of this fundamental problem. Together with our hardness results for $Q||C_{\max}$ this implies that the complexity of $P|HM|C_{\max}$ is the only remaining open case.

中文翻译:

在相关和不相关的机器上调度少数类型的作业的复杂性

在最小化总完工时间、加权完成时间的总和或负载向量的范数的同时,将作业调度到机器的任务是组合优化中最古老和最基本的任务之一。由于所有这些问题通常都是 NP-hard 问题,因此非常关注只有少量工作类型 $k$,但可能工作数量 $n$ 很大的制度;这是少数工作类型,高度多元化的制度。尽管有许多积极的结果,但直到现在才了解该制度的硬度边界。我们表明,在统一相关的机器 ($Q|HM|C_{\max}$) 上的 makespan 最小化已经是 NP-hard 的,具有 $6$ 的作业类型,并且相关的切割库存问题已经是 NP-hard 的,已经具有 $8$ 的项目类型. 对于更一般的无关机器模型 ($R|HM|C_{\max}$),我们表明,如果最大作业大小 $p_{\max}$ 或作业数量 $n$ 以多项式有界于实例大小 $|I|$,则存在复杂度为 $|I|^{\ textrm{poly}(k)}$。我们的主要结果是这不太可能得到改善,因为 $Q||C_{\max}$ 是 W[1]-hard 参数化的 $k$,当 $n$、$p_{\max}$ 和描述速度的数字是 $|I|$ 中的多项式;当作业规模矩阵的等级为 $2$ 时,$R|HM|C_{\max}$(没有速度)也是如此。我们的正面和负面结果也扩展到负载向量的 $\ell_2$-norm 最小化目标,部分是加权完成时间的总和 $\sum w_j C_j$。在此过程中,我们肯定地回答了相同机器上的完工时间最小化 ($P||C_{\max}$) 是否是由 $k$ 参数化的固定参数易处理的问题,扩展我们对这个基本问题的理解。连同我们对 $Q||C_{\max}$ 的硬度结果,这意味着 $P|HM|C_{\max}$ 的复杂性是唯一剩下的开放案例。
更新日期:2020-09-25
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