Computing Knapsack Solutions with Cardinality Robustness

  • Naonori Kakimura
  • Kazuhisa Makino
  • Kento Seimi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


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 ε > 0, the problem of deciding whether the knapsack problem admits a (ν + ε)-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 [7], 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.


Knapsack Problem Robust Solution Fully Polynomial Time Approximation Scheme Greedy Solution Independence System 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Naonori Kakimura
    • 1
  • Kazuhisa Makino
    • 1
  • Kento Seimi
    • 1
  1. 1.Department of Mathematical InformaticsUniversity of TokyoTokyoJapan

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