### Abstract

We study the problem of allocating a set of indivisible items among agents with additive valuations, with the goal of maximizing the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NP-hard, and our main result is the first efficient constant-factor approximation algorithm for this objective. We first observe that the integrality gap of the natural fractional relaxation is exponential, so we propose a different fractional allocation which implies a tighter upper bound and, after appropriate rounding, yields a good integral allocation. An interesting contribution of this work is the fractional allocation that we use. The relaxation of our problem can be solved efficiently using the Eisenberg-Gale program, whose optimal solution can be interpreted as a market equilibrium with the dual variables playing the role of item prices. Using this market-based interpretation, we define an alternative equilibrium allocation where the amount of spending that can go into any given item is bounded, thus keeping the highly priced items under-allocated, and forcing the agents to spend on lower priced items. The resulting equilibrium prices reveal more information regarding how to assign items so as to obtain a good integral allocation.

Original language | English (US) |
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Title of host publication | STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 371-380 |

Number of pages | 10 |

ISBN (Electronic) | 9781450335362 |

DOIs | |

State | Published - Jun 14 2015 |

Event | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 - Portland, United States Duration: Jun 14 2015 → Jun 17 2015 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | 14-17-June-2015 |

ISSN (Print) | 0737-8017 |

### Other

Other | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 |
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Country | United States |

City | Portland |

Period | 6/14/15 → 6/17/15 |

### Fingerprint

### Keywords

- Approximation Algorithms
- Fair Division
- Geometric Mean
- Nash Bargaining
- Nash Product
- Nash Social Welfare

### ASJC Scopus subject areas

- Software

### Cite this

*STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing*(pp. 371-380). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. 14-17-June-2015). Association for Computing Machinery. https://doi.org/10.1145/2746539.2746589