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Generalized assignment and knapsack problems in the random-order model.

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Τίτλος: Generalized assignment and knapsack problems in the random-order model.
Συγγραφείς: Klimm, Max1 (AUTHOR) klimm@math.tu-berlin.de, Knaack, Martin1 (AUTHOR) knaack@math.tu-berlin.de
Πηγή: Mathematical Programming. Aug2026, p1-24. 24p.
Θεματικοί όροι: *Packing problem (Mathematics), Assignment problems (Programming), Knapsack problems, Optimization algorithms, Online algorithms
Περίληψη: We study different online optimization problems in the random-order model. There is a finite set of bins with known capacities and a finite set of items arriving in uniform random order. Upon arrival of an item, its size and its value for each of the bins is revealed and it has to be decided immediately and irrevocably to which bin the item is assigned, or whether the item is rejected. In this setting, an algorithm is α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-competitive if the total expected value of all items assigned to the bins is at least an α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-fraction of the total value of an optimal assignment that knows all items beforehand. We give an algorithm that is α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-competitive with α=(1-ln2)/2≈1/6.52\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha = (1-\ln 2 )/2 \approx 1/6.52$$\end{document} improving upon the previous best algorithm with α≈1/6.99\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \approx 1/6.99$$\end{document} for the generalized assignment problem and the previous best algorithm with α≈1/6.65\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \approx 1/6.65$$\end{document} for the integral knapsack problem. We then study the fractional knapsack problem where we have a single bin and it is also allowed to pack items fractionally. For that case, we obtain an algorithm that is α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-competitive with α=1/e≈1/2.71\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha = 1/e \approx 1/2.71$$\end{document} improving on the previous best algorithm with α=1/4.39\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha = 1/4.39$$\end{document}. We further show that the competitive ratio of 1/e is best-possible for randomized algorithms in this model. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  – Url: https://dx.doi.org/doi:10.1007/s10107-026-02405-6
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  Data: Generalized assignment and knapsack problems in the random-order model.
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  Data: <searchLink fieldCode="AR" term="%22Klimm%2C+Max%22">Klimm, Max</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> klimm@math.tu-berlin.de</i><br /><searchLink fieldCode="AR" term="%22Knaack%2C+Martin%22">Knaack, Martin</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> knaack@math.tu-berlin.de</i>
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  Data: <searchLink fieldCode="JN" term="%22Mathematical+Programming%22">Mathematical Programming</searchLink>. Aug2026, p1-24. 24p.
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  Data: *<searchLink fieldCode="DE" term="%22Packing+problem+%28Mathematics%29%22">Packing problem (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Assignment+problems+%28Programming%29%22">Assignment problems (Programming)</searchLink><br /><searchLink fieldCode="DE" term="%22Knapsack+problems%22">Knapsack problems</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Online+algorithms%22">Online algorithms</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We study different online optimization problems in the random-order model. There is a finite set of bins with known capacities and a finite set of items arriving in uniform random order. Upon arrival of an item, its size and its value for each of the bins is revealed and it has to be decided immediately and irrevocably to which bin the item is assigned, or whether the item is rejected. In this setting, an algorithm is α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-competitive if the total expected value of all items assigned to the bins is at least an α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-fraction of the total value of an optimal assignment that knows all items beforehand. We give an algorithm that is α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-competitive with α=(1-ln2)/2≈1/6.52\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha = (1-\ln 2 )/2 \approx 1/6.52$$\end{document} improving upon the previous best algorithm with α≈1/6.99\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \approx 1/6.99$$\end{document} for the generalized assignment problem and the previous best algorithm with α≈1/6.65\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \approx 1/6.65$$\end{document} for the integral knapsack problem. We then study the fractional knapsack problem where we have a single bin and it is also allowed to pack items fractionally. For that case, we obtain an algorithm that is α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document}-competitive with α=1/e≈1/2.71\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha = 1/e \approx 1/2.71$$\end{document} improving on the previous best algorithm with α=1/4.39\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha = 1/4.39$$\end{document}. We further show that the competitive ratio of 1/<italic>e</italic> is best-possible for randomized algorithms in this model. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Mathematical Programming is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1007/s10107-026-02405-6
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      – Code: eng
        Text: English
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      Pagination:
        PageCount: 24
        StartPage: 1
    Subjects:
      – SubjectFull: Packing problem (Mathematics)
        Type: general
      – SubjectFull: Assignment problems (Programming)
        Type: general
      – SubjectFull: Knapsack problems
        Type: general
      – SubjectFull: Optimization algorithms
        Type: general
      – SubjectFull: Online algorithms
        Type: general
    Titles:
      – TitleFull: Generalized assignment and knapsack problems in the random-order model.
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            NameFull: Klimm, Max
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            NameFull: Knaack, Martin
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            – D: 12
              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
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            – TitleFull: Mathematical Programming
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