## 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 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, 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.

Original language | English |
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Pages (from-to) | 469-483 |

Number of pages | 15 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Oct |

Externally published | Yes |

## Keywords

- FPTAS
- Knapsack problems
- NP-hardness
- Robustness

## ASJC Scopus subject areas

- General Engineering
- Applied Mathematics