A Note on Approximating Weighted Nash Social Welfare with Additive Valuations

Authors Yuda Feng, Shi Li



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Author Details

Yuda Feng
  • Department of Computer Science and Technology, Harbin Institute of Technology, Heilongjiang, China
Shi Li
  • Department of Computer Science and Technology, Nanjing University, Jiangsu, China

Acknowledgements

YF is an incoming PhD student at Nanjing University, and his work was a part of his undergraduate dissertation supervised by SL.

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Yuda Feng and Shi Li. A Note on Approximating Weighted Nash Social Welfare with Additive Valuations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 63:1-63:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.63

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} + ε ≈ 1.445 + ε, which matches the best known approximation ratio for the unweighted case [Barman et al., 2018]. 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 [Shmoys and Tardos, 1993]. 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} ≈ 1.445 by Barman et al. [Barman et al., 2018], leading to our approximation ratio.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Nash Social Welfare
  • Configuration LP
  • Approximation Algorithms

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