A Quasi-Polynomial Approximation for the Restricted Assignment Problem

Scheduling jobs on unrelated machines and minimizing the makespan is a classical problem in combinatorial optimization. A job j has a processing time $$p_{ij}$$ for every machine i. The best polynomial algorithm known for this problem goes back to Lenstra et al. and has an approximation ratio of 2....

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Bibliographic Details
Published inInteger Programming and Combinatorial Optimization Vol. 10328; pp. 305 - 316
Main Authors Jansen, Klaus, Rohwedder, Lars
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 01.01.2017
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Approximation
Local search
Scheduling
Unrelated machines
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ISBN9783319592497
3319592491
ISSN0302-9743
1611-3349
DOI10.1007/978-3-319-59250-3_25

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Summary:Scheduling jobs on unrelated machines and minimizing the makespan is a classical problem in combinatorial optimization. A job j has a processing time $$p_{ij}$$ for every machine i. The best polynomial algorithm known for this problem goes back to Lenstra et al. and has an approximation ratio of 2. In this paper we study the Restricted Assignment problem, which is the special case where $$p_{ij}\in \{p_j,\infty \}$$ . We present an algorithm for this problem with an approximation ratio of $$11/6 + \epsilon $$ and quasi-polynomial running time $$n^{\mathcal O(1/\epsilon \log (n))}$$ for every $$\epsilon > 0$$ . This closes the gap to the best estimation algorithm known for the problem with regard to quasi-polynomial running time.
Bibliography:Research supported by German Research Foundation (DFG) project JA 612/15-1.
Original Abstract: Scheduling jobs on unrelated machines and minimizing the makespan is a classical problem in combinatorial optimization. A job j has a processing time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} for every machine i. The best polynomial algorithm known for this problem goes back to Lenstra et al. and has an approximation ratio of 2. In this paper we study the Restricted Assignment problem, which is the special case where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}\in \{p_j,\infty \}$$\end{document}. We present an algorithm for this problem with an approximation ratio of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$11/6 + \epsilon $$\end{document} and quasi-polynomial running time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{\mathcal O(1/\epsilon \log (n))}$$\end{document} for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon > 0$$\end{document}. This closes the gap to the best estimation algorithm known for the problem with regard to quasi-polynomial running time.
ISBN:9783319592497
3319592491
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-319-59250-3_25