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Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations

Authors Siddharth Barman, Umang Bhaskar, Anand Krishna, Ranjani G. Sundaram

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Author Details

Siddharth Barman
  • Indian Institute of Science, Bangalore, India
Umang Bhaskar
  • Tata Institute of Fundamental Research, Mumbai, India
Anand Krishna
  • Indian Institute of Science, Bangalore, India
Ranjani G. Sundaram
  • Chennai Mathematical Institute, India

Funding

  • Barman, Siddharth: Supported by a Ramanujan Fellowship (SERB - SB/S2/RJN128/2015) and a Pratiksha Trust Young Investigator Award.
  • Bhaskar, Umang: Supported by the Department of Atomic Energy, Government of India (project RTI4001), a Ramanujan Fellowship (SERB - SB/S2/RJN-055/2015), and an Early Career Research Award (SERB - ECR 002766).

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Siddharth Barman, Umang Bhaskar, Anand Krishna, and Ranjani G. Sundaram. Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.11

Abstract

We develop polynomial-time algorithms for the fair and efficient allocation of indivisible goods among n agents that have subadditive valuations over the goods. We first consider the Nash social welfare as our objective and design a polynomial-time algorithm that, in the value oracle model, finds an 8n-approximation to the Nash optimal allocation. Subadditive valuations include XOS (fractionally subadditive) and submodular valuations as special cases. Our result, even for the special case of submodular valuations, improves upon the previously best known O(n log n)-approximation ratio of Garg et al. (2020). 
More generally, we study maximization of p-mean welfare. The p-mean welfare is parameterized by an exponent term p ∈ (-∞, 1] and encompasses a range of welfare functions, such as social welfare (p = 1), Nash social welfare (p → 0), and egalitarian welfare (p → -∞). We give an algorithm that, for subadditive valuations and any given p ∈ (-∞, 1], computes (in the value oracle model and in polynomial time) an allocation with p-mean welfare at least 1/(8n) times the optimal. 
Further, we show that our approximation guarantees are essentially tight for XOS and, hence, subadditive valuations. We adapt a result of Dobzinski et al. (2010) to show that, under XOS valuations, an O (n^{1-ε}) approximation for the p-mean welfare for any p ∈ (-∞,1] (including the Nash social welfare) requires exponentially many value queries; here, ε > 0 is any fixed constant.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic game theory
Keywords
  • Discrete Fair Division
  • Nash Social Welfare
  • Subadditive Valuations
  • Submodular Valuations

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References

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