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 \(O(n\log n)\) and \(O(n^2)\) 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 \(O(n\log ^2 n)\) 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.
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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.
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Halman, N., Kovalyov, M.Y. & Quilliot, A. Max–max, max–min, min–max and min–min knapsack problems with a parametric constraint. 4OR-Q J Oper Res 21, 235–246 (2023). https://doi.org/10.1007/s10288-022-00509-1
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DOI: https://doi.org/10.1007/s10288-022-00509-1