### 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 [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.

Original language | English |
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Title of host publication | Algorithms and Computation - 22nd International Symposium, ISAAC 2011, Proceedings |

Pages | 693-702 |

Number of pages | 10 |

Volume | 7074 LNCS |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

Event | 22nd International Symposium on Algorithms and Computation, ISAAC 2011 - Yokohama, Japan Duration: 2011 Dec 5 → 2011 Dec 8 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7074 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 22nd International Symposium on Algorithms and Computation, ISAAC 2011 |
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Country | Japan |

City | Yokohama |

Period | 11/12/5 → 11/12/8 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Computation - 22nd International Symposium, ISAAC 2011, Proceedings*(Vol. 7074 LNCS, pp. 693-702). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7074 LNCS). https://doi.org/10.1007/978-3-642-25591-5_71

**Computing knapsack solutions with cardinality robustness.** / Kakimura, Naonori; Makino, Kazuhisa; Seimi, Kento.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Computation - 22nd International Symposium, ISAAC 2011, Proceedings.*vol. 7074 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 7074 LNCS, pp. 693-702, 22nd International Symposium on Algorithms and Computation, ISAAC 2011, Yokohama, Japan, 11/12/5. https://doi.org/10.1007/978-3-642-25591-5_71

}

TY - GEN

T1 - Computing knapsack solutions with cardinality robustness

AU - Kakimura, Naonori

AU - Makino, Kazuhisa

AU - Seimi, Kento

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84055190788&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84055190788&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-25591-5_71

DO - 10.1007/978-3-642-25591-5_71

M3 - Conference contribution

AN - SCOPUS:84055190788

SN - 9783642255908

VL - 7074 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 693

EP - 702

BT - Algorithms and Computation - 22nd International Symposium, ISAAC 2011, Proceedings

ER -