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

Author

Listed:
  • Nir Halman

    (Bar-Ilan University)

  • Mikhail Y. Kovalyov

    (National Academy of Sciences of Belarus)

  • Alain Quilliot

    (Université Blaise Pascal)

Abstract

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 $$O(n\log n)$$ O ( n log n ) and $$O(n^2)$$ O ( n 2 ) 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 $$O(n\log ^2 n)$$ O ( n log 2 n ) 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.

Suggested Citation

  • Nir Halman & Mikhail Y. Kovalyov & Alain Quilliot, 2023. "Max–max, max–min, min–max and min–min knapsack problems with a parametric constraint," 4OR, Springer, vol. 21(2), pages 235-246, June.
    • Handle: RePEc:spr:aqjoor:v:21:y:2023:i:2:d:10.1007_s10288-022-00509-1
    DOI: 10.1007/s10288-022-00509-1
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    References listed on IDEAS

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    1. Maria Scutellà, 2007. "A note on the parametric maximum flow problem and some related reoptimization issues," Annals of Operations Research, Springer, vol. 150(1), pages 231-244, March.
    2. S. Thomas McCormick, 1999. "Fast Algorithms for Parametric Scheduling Come From Extensions to Parametric Maximum Flow," Operations Research, INFORMS, vol. 47(5), pages 744-756, October.
    3. Moshe Eben-Chaime, 1996. "Parametric Solution for Linear Bicriteria Knapsack Models," Management Science, INFORMS, vol. 42(11), pages 1565-1575, November.
    4. Nimrod Megiddo, 1979. "Combinatorial Optimization with Rational Objective Functions," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 414-424, November.
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