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Online interval scheduling on two related machines: the power of lookahead

Authors
  • Pinson, Nicolas1
  • Spieksma, Frits C. R.2
  • 1 ENS-Lyon, Lyon, France , Lyon (France)
  • 2 Eindhoven University of Technology, Department of Mathematics and Computer Science, Eindhoven, The Netherlands , Eindhoven (Netherlands)
Type
Published Article
Journal
Journal of Combinatorial Optimization
Publisher
Springer-Verlag
Publication Date
Jan 09, 2019
Volume
38
Issue
1
Pages
224–253
Identifiers
DOI: 10.1007/s10878-019-00381-6
Source
Springer Nature
Keywords
License
Green

Abstract

We consider an online interval scheduling problem on two related machines. If one machine is at least as twice as fast as the other machine, we say the machines are distinct; otherwise the machines are said to be similar. Each job j∈J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in J$$\end{document} is characterized by a length pj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_j$$\end{document}, and an arrival time tj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_j$$\end{document}; the question is to determine whether there exists a feasible schedule such that each job starts processing at its arrival time. For the case of unit-length jobs, we prove that when the two machines are distinct, there is an amount of lookahead allowing an online algorithm to solve the problem. When the two machines are similar, we show that no finite amount of lookahead is sufficient to solve the problem in an online fashion. We extend these results to jobs having arbitrary lengths, and consider an extension focused on minimizing total waiting time.

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