### 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 |

Volume | 14-17-June-2015 |

ISBN (Print) | 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 |

### Other

Other | 47th Annual ACM Symposium on Theory of Computing, STOC 2015 |
---|---|

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*(Vol. 14-17-June-2015, pp. 371-380). Association for Computing Machinery. https://doi.org/10.1145/2746539.2746589

**Approximating the Nash social welfare with indivisible items.** / Cole, Richard; Gkatzelis, Vasilis.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing.*vol. 14-17-June-2015, Association for Computing Machinery, pp. 371-380, 47th Annual ACM Symposium on Theory of Computing, STOC 2015, Portland, United States, 6/14/15. https://doi.org/10.1145/2746539.2746589

}

TY - GEN

T1 - Approximating the Nash social welfare with indivisible items

AU - Cole, Richard

AU - Gkatzelis, Vasilis

PY - 2015/6/14

Y1 - 2015/6/14

N2 - 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.

AB - 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.

KW - Approximation Algorithms

KW - Fair Division

KW - Geometric Mean

KW - Nash Bargaining

KW - Nash Product

KW - Nash Social Welfare

UR - http://www.scopus.com/inward/record.url?scp=84958775370&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958775370&partnerID=8YFLogxK

U2 - 10.1145/2746539.2746589

DO - 10.1145/2746539.2746589

M3 - Conference contribution

SN - 9781450335362

VL - 14-17-June-2015

SP - 371

EP - 380

BT - STOC 2015 - Proceedings of the 2015 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

ER -