## Abstract

In this paper, we investigate the parametric weight knapsack problem, in which the item weights are affine functions of the form w_{i}(λ)=a_{i}+λ⋅b_{i} for i∈{1,…,n} depending on a real-valued parameter λ. The aim is to provide a solution for all values of the parameter. It is well-known that any exact algorithm for the problem may need to output an exponential number of knapsack solutions. We present the first fully polynomial-time approximation schemes (FPTASs) for the problem that, for any desired precision ε∈(0,1), compute (1−ε)-approximate solutions for all values of the parameter. Among others, we present a strongly polynomial FPTAS running in O[Formula presented]⋅log^{3}[Formula presented]logn time and a weakly polynomial FPTAS with a running time of O[Formula presented]⋅log^{2}[Formula presented]⋅logP⋅logMlogn, where P is an upper bound on the optimal profit and M≔max{|W|,n⋅max{|a_{i}|,|b_{i}|:i∈{1,…,n}}} for a knapsack with capacity W.

Original language | English |
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Pages (from-to) | 487-491 |

Number of pages | 5 |

Journal | Operations Research Letters |

Volume | 46 |

Issue number | 5 |

DOIs | |

State | Published - Sep 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier B.V.

## Keywords

- Approximation algorithms
- Knapsack problems
- Parametric optimization