## Abstract

In classic bin packing, the objective is to partition a set of n items with positive rational sizes in (0, 1] into a minimum number of subsets called bins, such that the total size of the items of each bin at most 1. We study a bin packing game where the cost of each bin is 1, and given a valid packing of the items, each item has a cost associated with it, such that the items that are packed into a bin share its cost equally. We find tight bounds on the exact worst-case number of steps in processes of convergence to pure Nash equilibria. Those are processes that are given an arbitrary packing as an initial packing. As long as there exists an item that can reduce its cost by moving from its bin to another bin, in each step, a controller selects such an item and instructs it to perform such a beneficial move. The process converges when no further beneficial moves exist. The tight function of n that we find is in Θ(n^{3}/^{2}). This improves the previous bound of Ma et al. [14], who showed an upper bound of O(n^{2}).

Original language | English |
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Pages (from-to) | 853-865 |

Number of pages | 13 |

Journal | Acta Cybernetica |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Funding Information:∗Supported by VKSZ 12-1-2013-0088 “Development of cloud based smart IT solutions by IBM Hungary in cooperation with the University of Pannonia” and by National Research, Development and Innovation Office – NKFIH under the grant SNN 116095. aDepartment of Mathematics, University of Pannonia, Veszprém, Hungary, E-mail: dosagy@almos.vein.hu bDepartment of Mathematics, University of Haifa, Haifa, Israel. lea@math.haifa.ac.il

Publisher Copyright:

Copyright © 2018 Institute of Informatics, University of Szeged. All rights reserved.

## ASJC Scopus subject areas

- Software
- Computer Science (miscellaneous)
- Computer Vision and Pattern Recognition
- Management Science and Operations Research
- Information Systems and Management
- Electrical and Electronic Engineering