## Abstract

We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ε h for some suitable parameter h and any given ε > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs.

Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ε⋅ p_{max}.

Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ε⋅ p_{max}.

Original language | English |
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Title of host publication | Proceedings of 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) |

Editors | Nikhil Bansal, Emanuela Merelli, James Worrell |

Place of Publication | Saarbrücken/Wadern, Germany |

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

Pages | 42:1-42:17 |

Volume | 198 |

Edition | LIPIcs |

ISBN (Print) | 978-3-95977-195-5 |

DOIs | |

Publication status | Published - 2021 |

### Publication series

Series | Leibniz International Proceedings in Informatics (LIPIcs) |
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Volume | 198 |

ISSN | 1868-8969 |

## Keywords

- load balancing
- approximation schemes
- parallel machine scheduling