Max–max, max–min, min–max and min–min knapsack problems with a parametric constraint

Nir Halman, Mikhail Y. Kovalyov, Alain Quilliot

Research output: Contribution to journalArticlepeer-review


Max–max, max–min, min–max and min–min optimization problems with a knapsack-type constraint containing a single numerical parameter are studied. The goal is to present optimal solutions for all possible values of the parameter. Algorithms with O(nlog n) and O(n2) running times are proposed for the problems with a fixed parameter and for the general problem, respectively, where n is the number of items to be packed into the knapsack. The latter algorithm determines optimal solution values for all values of the parameter in O(nlog 2n) time. The problem of deciding whether there exists a single optimal solution for all values of the numerical parameter is proved to be NP-complete.

Original languageEnglish
Pages (from-to)235-246
Number of pages12
Issue number2
StatePublished - 2022

Bibliographical note

Funding Information:
Nir Halman is supported in part by the Israel Science Foundation grants 399/17 and 1074/21, and by the United States-Israel Binational Science Foundation (BSF). Mikhail Y. Kovalyov and Alain Quilliot are supported in part by French ANR, Labex IMOBS3, and PGMO Program.

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.


  • Knapsack problems
  • Parametric optimization
  • Polynomial algorithm


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