## Abstract

We consider robust variants of the bin packing problem with uncertain item sizes. Specifically we consider two uncertainty sets previously studied in the literature. The first is budgeted uncertainty (the U^{Γ} model), in which at most Γ items deviate, each reaching its peak value, while other items assume their nominal values. The second uncertainty set, the U^{Ω} model, bounds the total amount of deviation in each scenario. We show that a variant of the Next-cover algorithm is a 2 approximation for the U^{Ω} model, and another variant of this algorithm is a 2Γ approximation for the U^{Γ} model. Unlike the classical bin packing problem, it is shown that (unless P = N P ) no asymptotic approximation scheme exists for the U^{Γ} model, for Γ “1. This motivates the question of the existence of a constant approximation factor algorithm for the U^{Γ} model. Our main result is to answer this question by proving a (polynomial-time) 4.5 approximation algorithm, based on a dynamic-programming approach.

Original language | English |
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Pages (from-to) | 2534-2552 |

Number of pages | 19 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 36 |

Issue number | 4 |

DOIs | |

State | Published - 2022 |

### Bibliographical note

Funding Information:˚Received by the editors November 3, 2021; accepted for publication (in revised form) June 5, 2022; published electronically October 24, 2022. A preliminary version of this work has been published in [7]. https://doi.org/10.1137/21M1457199 Funding: This research has benefitted from the support of the ANR project ROBUST (ANR-16-CE40-0018). :LIRMM, University of Montpellier, CNRS, Montpellier 34095, France (marin.bougeret@lirmm.fr, michael.poss@lirmm.fr). ;Department of Mathematics, University of Pannonia, Veszpr\e'm 8200, Hungary (dosagy@almos. vein.hu). \S Department of Management, Bar-Ilan University, Ramat Gan 5290002, Israel (noam.goldberg@ biu.ac.il).

Publisher Copyright:

© 2022 Society for Industrial and Applied Mathematics.

## Keywords

- Next-cover
- approximation algorithms
- bin-packing
- dynamic programming
- robust optimization