Abstract
In this paper, we study the robustness over the cardinality variation for the knapsack problem. For the knapsack problem and a positive number α ≤ 1, we say that a feasible solution is α-robust if, for any positive integer k, it includes an α-approximation of the maximum k-knapsack solution, where a k-knapsack solution is a feasible solution that consists of at most k items. In this paper, we show that, for any \({\varepsilon >0 }\) , the problem of deciding whether the knapsack problem admits a \({(\nu+\varepsilon)}\) -robust solution is weakly NP-hard, where ν denotes the rank quotient of the corresponding knapsack system. Since the knapsack problem always admits a ν-robust knapsack solution, this result provides a sharp border for the complexity of the robust knapsack problem. On the positive side, we show that a max-robust knapsack solution can be computed in pseudo-polynomial time, and present a fully polynomial-time approximation scheme (FPTAS) for computing a max-robust knapsack solution.
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A preliminary version appears in Proceedings of the 22nd International Symposium on Algorithms and Computation (ISAAC 2011) [14].
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Kakimura, N., Makino, K. & Seimi, K. Computing knapsack solutions with cardinality robustness. Japan J. Indust. Appl. Math. 29, 469–483 (2012). https://doi.org/10.1007/s13160-012-0075-z
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DOI: https://doi.org/10.1007/s13160-012-0075-z