### Abstract

We consider a natural generalization of the classical multiple knapsack problem in which instead of packing single items we are packing groups of items. In this problem, we have multiple knapsacks and a set of items which are partitioned into groups. Each item has an individual weight, while the profit is associated with groups rather than items. The profit of a group can be attained if and only if every item of this group is packed. Such a general model finds applications in various practical problems, e.g., delivering bundles of goods. The tractability of this problem relies heavily on how large a group could be. Deciding if a group of items of total weight 2 could be packed into two knapsacks of unit capacity is already NP-hard and it thus rules out a constantapproximation algorithm for this problem in general. We then focus on the parameterized version where the total weight of items in each group is bounded by a factor δ of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to δ are derived. We also show that, depending on whether the number of knapsacks is a constant or part of the input, the approximation ratio for the problem, as a function on δ, changes substantially, which has a clear difference from the classical multiple knapsack problem.

Original language | English |
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Title of host publication | 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016 |

Editors | Heribert Vollmer, Nicolas Ollinger |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770019 |

DOIs | |

State | Published - Feb 1 2016 |

Event | 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016 - Orleans, France Duration: Feb 17 2016 → Feb 20 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 47 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016 |
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Country | France |

City | Orleans |

Period | 02/17/16 → 02/20/16 |

### Keywords

- Approximation algorithms
- Bin packing
- Lower bound
- Multiple knapsack

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## Cite this

*33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016*[28] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 47). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.STACS.2016.28