## Abstract

We consider robust variants of the bin-packing problem where the sizes of the items can take any value in a given uncertainty set (formula presented), where (formula presented) represents the nominal sizes of the items and (formula presented) their possible deviations. We consider more specifically two uncertainty sets previously studied in the literature. The first set, denoted U^{Γ}, contains scenarios in which at most Γ∈ N items deviate, each of them reaching its peak value (formula presented), while each other item has its nominal value (formula presented). The second set, denoted U^{Ω}, bounds by Ω∈ [ 0, 1 ] the total amount of deviation in each scenario. We show that a variant of the next-fit algorithm provides a 2-approximation for model U^{Ω}, and a 2 (Γ+ 1) approximation for model U^{Γ} (which can be improved to 2 approximation for Γ= 1). This motivates the question of the existence of a constant ratio approximation algorithm for the U^{Γ} model. Our main result is to answer positively to this question by providing a 4.5 approximation for U^{Γ} model based on dynamic programming.

Original language | English |
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Title of host publication | Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings |

Editors | Zachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack |

Publisher | Springer Verlag |

Pages | 71-84 |

Number of pages | 14 |

ISBN (Print) | 9783030247652 |

DOIs | |

State | Published - 2019 |

Event | 16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada Duration: 5 Aug 2019 → 7 Aug 2019 |

### 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 | 11646 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 16th International Symposium on Algorithms and Data Structures, WADS 2019 |
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Country/Territory | Canada |

City | Edmonton |

Period | 5/08/19 → 7/08/19 |

### Bibliographical note

Publisher Copyright:© Springer Nature Switzerland AG 2019.

## Keywords

- Approximation algorithm
- Bin-packing
- Dynamic programming
- Next-fit
- Robust optimization