2000 character limit reached
A Note on Approximating Weighted Nash Social Welfare with Additive Valuations
Published 24 Apr 2024 in cs.GT and cs.DS | (2404.15607v2)
Abstract: We give the first $O(1)$-approximation for the weighted Nash Social Welfare problem with additive valuations. The approximation ratio we obtain is $e{1/e} + \epsilon \approx 1.445 + \epsilon$, which matches the best known approximation ratio for the unweighted case \cite{BKV18}. Both our algorithm and analysis are simple. We solve a natural configuration LP for the problem, and obtain the allocation of items to agents using a randomized version of the Shmoys-Tardos rounding algorithm developed for unrelated machine scheduling problems. In the analysis, we show that the approximation ratio of the algorithm is at most the worst gap between the Nash social welfare of the optimum allocation and that of an EF1 allocation, for an unweighted Nash Social Welfare instance with identical additive valuations. This was shown to be at most $e{1/e} \approx 1.445$ by Barman et al., leading to our approximation ratio.
- Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 36:1–36:12, Dagstuhl, Germany, 2017. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
- The Geometry of Efficient Fair Division. The Geometry of Efficient Fair Division, pages 1–462, 01 2005.
- Finding Fair and Efficient Allocations. In Proceedings of the 2018 ACM Conference on Economics and Computation, EC ’18, page 557–574, New York, NY, USA, 2018. Association for Computing Machinery.
- Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press, 1996.
- Handbook of Computational Social Choice. Cambridge University Press, USA, 1st edition, 2016.
- Approximation Algorithms for the Weighted Nash Social Welfare via Convex and Non-Convex Programs. In Proceedings of the Thirty-Fifth ACM-SIAM Symposium on Discrete Algorithms, 2024.
- Bargaining Among Groups: An Axiomatic Viewpoint. International Journal of Game Theory, 39(1):71–88, 2010.
- Convex Program Duality, Fisher Markets, and Nash Social Welfare. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC ’17, page 459–460, New York, NY, USA, 2017. Association for Computing Machinery.
- Approximating the Nash Social Welfare with Indivisible Items. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’15, page 371–380, New York, NY, USA, 2015. Association for Computing Machinery.
- Bankruptcy to Surplus: Sharing Transboundary River Basin’s Water under Scarcity. Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), 32(8):2735–2751, 2018.
- A Constant Factor Approximation for Nash Social Welfare with Subadditive Valuations. ArXiv, abs/2309.04656, 2023.
- Satiation in Fisher Markets and Approximation of Nash Social Welfare. Mathematics of Operations Research, 0(0):null, 2023.
- Approximating Nash Social Welfare by Matching and Local Search. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, page 1298–1310, New York, NY, USA, 2023. Association for Computing Machinery.
- Approximating Nash Social Welfare under Rado Valuations. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, page 1412–1425, New York, NY, USA, 2021. Association for Computing Machinery.
- Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings. ACM Trans. Algorithms, 19(4), sep 2023.
- A generalized nash solution for two-person bargaining games with incomplete information. Management Science, 18(5):P80–P106, 1972.
- Asymmetric Nash Solutions in the River Sharing Problem. Game Theory & Bargaining Theory eJournal, 2013.
- E. Kalai. Nonsymmetric Nash Solutions and Replications of 2-Person Bargaining. Int. J. Game Theory, 6(3):129–133, sep 1977.
- The Nash Social Welfare Function. Econometrica, 47(2):423–435, 1979.
- Frank Kelly. Charging and Rate Control for Elastic Traffic. European Transactions on Telecommunications, 8(1):33–37, 1997.
- Bargaining in Committees as An Extension of Nash’s Bargaining Theory. Journal of Economic Theory, 132(1):291–305, 2007.
- Euiwoong Lee. APX-Hardness of Maximizing Nash Social Welfare with Indivisible Items. ArXiv, abs/1507.01159, 2015.
- W. Li and J. Vondrak. A Constant-Factor Approximation Algorithm for Nash Social Welfare with Submodular Valuations. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 25–36, Los Alamitos, CA, USA, feb 2022. IEEE Computer Society.
- Estimating the Nash Social Welfare for Coverage and Other Submodular Valuations. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’21, page 1119–1130, USA, 2021. Society for Industrial and Applied Mathematics.
- Herve Moulin. Fair Division and Collective Welfare. The MIT Press, 2004.
- John F. Nash. The Bargaining Problem. Econometrica, 18(2):155–162, 1950.
- Complexity and Approximability of Social Welfare Optimization in Multiagent Resource Allocation. In Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 3, AAMAS ’12, page 1287–1288, Richland, SC, 2012. International Foundation for Autonomous Agents and Multiagent Systems.
- Cake-Cutting Algorithms: Be Fair if You Can. A K Peters/CRC Press, 1998.
- Jrg Rothe. Economics and Computation: An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division. Springer Publishing Company, Incorporated, 1st edition, 2015.
- An Approximation Algorithm for the Generalized Assignment Problem. Mathematical Programming, 62(1):461–474, 1993.
- William Thomson. Replication Invariance of Bargaining Solutions. Int. J. Game Theory, 15(1):59–63, mar 1986.
- Hal R Varian. Equity, Envy, and Efficiency. Journal of Economic Theory, 9(1):63–91, 1974.
- H. Peyton Young. Equity: In Theory and Practice. Princeton University Press, 1994.
- Nash Bargaining Solutions for International Climate Agreements under Different Sets of Bargaining Weights. International Environmental Agreements: Politics, Law and Economics, 17(5):709–729, 2017.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.