research-article

Approximating the Nash Social Welfare with Indivisible Items

Authors:

Vasilis GkatzelisAuthors Info & Claims

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

Pages 371 - 380

https://doi.org/10.1145/2746539.2746589

Published: 14 June 2015 Publication History

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.

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Cited By

Wu XLi BGan J(2025)Approximate envy-freeness in indivisible resource allocation with budget constraintsInformation and Computation10.1016/j.ic.2024.105264(105264)Online publication date: Jan-2025
https://doi.org/10.1016/j.ic.2024.105264
Liao WJin YWang ZWang XXia X(2024)Maximizing Nash Social Welfare Based on Greedy Algorithm and Estimation of Distribution AlgorithmBiomimetics10.3390/biomimetics91106529:11(652)Online publication date: 24-Oct-2024
https://doi.org/10.3390/biomimetics9110652
Dai SGao GLiu SLim BNing LXu YZhang Y(2024)Maximum Nash Social Welfare Under Budget-Feasible EFXIEEE Transactions on Network Science and Engineering10.1109/TNSE.2023.333210411:2(1810-1820)Online publication date: Mar-2024
https://doi.org/10.1109/TNSE.2023.3332104
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Index Terms

Approximating the Nash Social Welfare with Indivisible Items
1. Theory of computation
  1. Randomness, geometry and discrete structures

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Approximating the Nash Social Welfare with Indivisible Items

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cover image ACM Conferences

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

June 2015

916 pages

ISBN:9781450335362

DOI:10.1145/2746539

General Chair:
Rocco Servedio
Columbia University
,
Program Chair:
Ronitt Rubinfeld
MIT and Tel Aviv University

Copyright © 2015 ACM.

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Published: 14 June 2015

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STOC '15: Symposium on Theory of Computing

June 14 - 17, 2015

Oregon, Portland, USA

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STOC '15 Paper Acceptance Rate 93 of 347 submissions, 27%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Wu XLi BGan J(2025)Approximate envy-freeness in indivisible resource allocation with budget constraintsInformation and Computation10.1016/j.ic.2024.105264(105264)Online publication date: Jan-2025
https://doi.org/10.1016/j.ic.2024.105264
Liao WJin YWang ZWang XXia X(2024)Maximizing Nash Social Welfare Based on Greedy Algorithm and Estimation of Distribution AlgorithmBiomimetics10.3390/biomimetics91106529:11(652)Online publication date: 24-Oct-2024
https://doi.org/10.3390/biomimetics9110652
Dai SGao GLiu SLim BNing LXu YZhang Y(2024)Maximum Nash Social Welfare Under Budget-Feasible EFXIEEE Transactions on Network Science and Engineering10.1109/TNSE.2023.333210411:2(1810-1820)Online publication date: Mar-2024
https://doi.org/10.1109/TNSE.2023.3332104
Samieefar K(2023)A Meta-heuristic Approach for Strategic Fair Division ProblemsProceedings of the 2023 7th International Conference on Intelligent Systems, Metaheuristics & Swarm Intelligence10.1145/3596947.3596969(87-94)Online publication date: 23-Apr-2023
https://dl.acm.org/doi/10.1145/3596947.3596969
Dobzinski SOren SVondrak JLeyton-Brown KSamuelson LHartline J(2023)Fairness and Incentive Compatibility via Percentage FeesProceedings of the 24th ACM Conference on Economics and Computation10.1145/3580507.3597810(517-535)Online publication date: 9-Jul-2023
https://dl.acm.org/doi/10.1145/3580507.3597810
Meskar ELiang B(2023)Fair Multi-Resource Allocation in Heterogeneous Servers With an External Resource TypeIEEE/ACM Transactions on Networking10.1109/TNET.2022.321342631:3(1244-1262)Online publication date: Jun-2023
https://doi.org/10.1109/TNET.2022.3213426
Garg JMurhekar A(2023)Computing fair and efficient allocations with few utility valuesTheoretical Computer Science10.1016/j.tcs.2023.113932962(113932)Online publication date: Jun-2023
https://doi.org/10.1016/j.tcs.2023.113932
Sun AChen BDoan X(2023)Fairness criteria for allocating indivisible chores: connections and efficienciesAutonomous Agents and Multi-Agent Systems10.1007/s10458-023-09618-537:2Online publication date: 31-Aug-2023
https://doi.org/10.1007/s10458-023-09618-5
Huang ZLi MShu XWei T(2023)Online Nash Welfare Maximization Without PredictionsWeb and Internet Economics10.1007/978-3-031-48974-7_23(402-419)Online publication date: 31-Dec-2023
https://doi.org/10.1007/978-3-031-48974-7_23
Dai SGao GLiu SLim BNing LXu YZhang Y(2023)EFX Under Budget ConstraintFrontiers of Algorithmic Wisdom10.1007/978-3-031-20796-9_1(3-14)Online publication date: 1-Jan-2023
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