Document Open Access Logo

Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations

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

PDF
Thumbnail PDF

File

LIPIcs.ESA.2020.11.pdf
  • Filesize: 0.58 MB
  • 17 pages

Document Identifiers

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).

Cite AsGet BibTex

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Nima Anari, Shayan Oveis Gharan, Tung Mai, and Vijay V Vazirani. Nash Social Welfare for Indivisible Items under Separable, Piecewise-Linear Concave Utilities. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2274-2290, 2018. Google Scholar
  2. Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In Proceedings of the 8th Conference on Innovations in Theoretical Computer Science (ITCS), 2017. Google Scholar
  3. Haris Aziz. Developments in multi-agent fair allocation. In AAAI, pages 13563-13568, 2020. Google Scholar
  4. Siddharth Barman, Umang Bhaskar, Anand Krishna, and Ranjani G. Sundaram. Tight approximation algorithms for p-mean welfare under subadditive valuations. CoRR, abs/2005.07370, 2020. URL: http://arxiv.org/abs/2005.07370.
  5. Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Finding fair and efficient allocations. In Proceedings of the 2018 ACM Conference on Economics and Computation, Ithaca, NY, USA, June 18-22, 2018, pages 557-574. ACM, 2018. Google Scholar
  6. Xiaohui Bei, Jugal Garg, Martin Hoefer, and Kurt Mehlhorn. Earning Limits in Fisher Markets with Spending-Constraint Utilities. In Proceedings of the International Symposium on Algorithmic Game Theory (SAGT), pages 67-79, 2017. Google Scholar
  7. Ivona Bezáková and Varsha Dani. Allocating indivisible goods. SIGecom Exchanges, 5(3):11-18, 2005. Google Scholar
  8. Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D Procaccia. Handbook of computational social choice. Cambridge University Press, 2016. Google Scholar
  9. Ioannis Caragiannis, David Kurokawa, Hervé Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang. The unreasonable fairness of maximum nash welfare. ACM Trans. Economics and Comput., 7(3):12:1-12:32, 2019. Google Scholar
  10. Deeparnab Chakrabarty, Julia Chuzhoy, and Sanjeev Khanna. On allocating goods to maximize fairness. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 107-116. IEEE Computer Society, 2009. Google Scholar
  11. Bhaskar Ray Chaudhury, Yun Kuen Cheung, Jugal Garg, Naveen Garg, Martin Hoefer, and Kurt Mehlhorn. On fair division for indivisible items. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2018, December 11-13, 2018, Ahmedabad, India, volume 122 of LIPIcs, pages 25:1-25:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  12. Bhaskar Ray Chaudhury, Jugal Garg, and Ruta Mehta. Fair and efficient allocations under subadditive valuations, 2020. URL: http://arxiv.org/abs/2005.06511.
  13. Richard Cole, Nikhil R. Devanur, Vasilis Gkatzelis, Kamal Jain, Tung Mai, Vijay V. Vazirani, and Sadra Yazdanbod. Convex program duality, fisher markets, and nash social welfare. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017, pages 459-460. ACM, 2017. Google Scholar
  14. Richard Cole and Vasilis Gkatzelis. Approximating the Nash Social Welfare with Indivisible Items. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), pages 371-380, 2015. Google Scholar
  15. Vincent Conitzer, Rupert Freeman, and Nisarg Shah. Fair public decision making. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017, pages 629-646. ACM, 2017. Google Scholar
  16. Shahar Dobzinski, Noam Nisan, and Michael Schapira. Approximation algorithms for combinatorial auctions with complement-free bidders. Math. Oper. Res., 35(1):1-13, 2010. Google Scholar
  17. Ulle Endriss. Trends in Computational Social Choice. Lulu. com, 2017. Google Scholar
  18. Uriel Feige. On maximizing welfare when utility functions are subadditive. SIAM J. Comput., 39(1):122-142, 2009. Google Scholar
  19. Jugal Garg, Martin Hoefer, and Kurt Mehlhorn. Approximating the Nash Social Welfare with Budget-Additive Valuations. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2326-2340, 2018. Google Scholar
  20. Jugal Garg, Pooja Kulkarni, and Rucha Kulkarni. Approximating nash social welfare under submodular valuations through (un)matchings. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2673-2687. SIAM, 2020. Google Scholar
  21. Michel X. Goemans, Nicholas J. A. Harvey, Satoru Iwata, and Vahab S. Mirrokni. Approximating submodular functions everywhere. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 535-544. SIAM, 2009. Google Scholar
  22. Jonathan Goldman and Ariel D Procaccia. Spliddit: Unleashing fair division algorithms. ACM SIGecom Exchanges, 13(2):41-46, 2015. Google Scholar
  23. Subhash Khot and Ashok Kumar Ponnuswami. Approximation algorithms for the max-min allocation problem. In APPROX-RANDOM, 2007. Google Scholar
  24. Euiwoong Lee. Apx-hardness of maximizing nash social welfare with indivisible items. Inf. Process. Lett., 122:17-20, 2017. Google Scholar
  25. Hervé Moulin. Fair division and collective welfare. MIT Press, 2003. Google Scholar
  26. Trung Thanh Nguyen and Jörg Rothe. Minimizing envy and maximizing average nash social welfare in the allocation of indivisible goods. Discret. Appl. Math., 179:54-68, 2014. Google Scholar
  27. Noam Nisan, T Roughgarden, E Tardos, and Vijay V Vazirani. Algorithmic game theory - Cambridge University Press, 2007. Google Scholar
  28. Zoya Svitkina and Lisa Fleischer. Submodular approximation: Sampling-based algorithms and lower bounds. SIAM Journal on Computing, 40(6):1715-1737, 2011. Google Scholar
  29. Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 67-74. ACM, 2008. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact