On the parameterized complexity of interval scheduling with eligible machine sets

IF 1.1 3区计算机科学 Q1 BUSINESS, FINANCE Journal of Computer and System Sciences Pub Date : 2024-03-29 DOI:10.1016/j.jcss.2024.103533

Danny Hermelin , Yuval Itzhaki , Hendrik Molter , Dvir Shabtay

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Abstract

We provide new parameterized complexity results for Interval Scheduling on Eligible Machines. In this problem, a set of n jobs is given to be processed non-preemptively on a set of m machines. Each job has a processing time, a deadline, a weight, and a set of eligible machines that can process it. The goal is to find a maximum weight subset of jobs that can each be processed on one of its eligible machines such that it completes exactly at its deadline. We focus on two parameters: The number m of machines, and the largest processing time $p_{\max}$ . Our main contribution is showing W[1]-hardness when parameterized by m. This answers Open Problem 8 of Mnich and van Bevern's list of 15 open problems in parameterized complexity of scheduling problems [Computers & Operations Research, 2018]. Furthermore, we show NP-hardness even when $p_{\max} = O (1)$ and present an FPT-algorithm with for the combined parameter $m + p_{\max}$ .

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论带合格机器集的区间调度的参数化复杂性

我们为合格机器上的间隔调度问题提供了新的参数化复杂度结果。在这个问题中，需要在一组 m 台机器上非抢占式地处理 n 个作业。每个作业都有一个处理时间、一个截止日期、一个权重和一组可以处理它的合格机器。我们的目标是找到一个最大权重的作业子集，每个作业都能在其中一台符合条件的机器上处理，从而在截止日期前完成作业。我们关注两个参数：机器数量 m 和最大处理时间 pmax。我们的主要贡献是证明了以 m 为参数时的 W[1]-hardness 性。这回答了 Mnich 和 van Bevern 列出的调度问题参数化复杂性 15 个开放问题中的开放问题 8 [Computers & Operations Research, 2018]。此外，我们还证明了即使 pmax=O(1) 时的 NP 难度，并提出了一种针对 m+pmax 组合参数的 FPT 算法。

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来源期刊

Journal of Computer and System Sciences 工程技术-计算机：理论方法

CiteScore

3.70

自引率

0.00%

发文量

审稿时长

68 days

期刊介绍： The Journal of Computer and System Sciences publishes original research papers in computer science and related subjects in system science, with attention to the relevant mathematical theory. Applications-oriented papers may also be accepted and they are expected to contain deep analytic evaluation of the proposed solutions. Research areas include traditional subjects such as: • Theory of algorithms and computability • Formal languages • Automata theory Contemporary subjects such as: • Complexity theory • Algorithmic Complexity • Parallel & distributed computing • Computer networks • Neural networks • Computational learning theory • Database theory & practice • Computer modeling of complex systems • Security and Privacy.