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Nash Social Welfare, Matrix Permanent, and Stable Polynomials

Authors Nima Anari, Shayan Oveis Gharan, Amin Saberi, Mohit Singh

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Keywords
  • Nash Welfare
  • Permanent
  • Matching
  • Stable Polynomial
  • Randomized Algorithm
  • Saddle Point

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Abstract

We study the problem of allocating m items to n agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a 1/e approximation factor of the objective, breaking the 1/2e^(1/e) approximation factor of Cole and Gkatzelis.

Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree m-homogeneous stable polynomial on m variables to all homogeneous polynomials. We  use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

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Nima Anari, Shayan Oveis Gharan, Amin Saberi, and Mohit Singh. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 36:1-36:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ITCS.2017.36

Author Details

Nima Anari
Shayan Oveis Gharan
Amin Saberi
Mohit Singh

References

  1. Arash Asadpour, Uriel Feige, and Amin Saberi. Santa claus meets hypergraph matchings. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 10-20. Springer, 2008. Google Scholar
  2. Arash Asadpour and Amin Saberi. An approximation algorithm for max-min fair allocation of indivisible goods. In STOC, pages 114-121, 2007. Google Scholar
  3. N Bansal and M Sviridenko. The santa claus problem. In Proceedings of Symposium on Theory of Computing, pages 31-40, 2006. Google Scholar
  4. S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2006. Google Scholar
  5. Ioannis Caragiannis, David Kurokawa, Hervé Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang. The unreasonable fairness of maximum nash welfare. In EC, 2016. Google Scholar
  6. Deeparnab Chakrabarty, Julia Chuzhoy, and Sanjeev Khanna. On allocating goods to maximize fairness. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 107-116. IEEE, 2009. Google Scholar
  7. 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. submitted, 2016. Google Scholar
  8. Richard Cole and Vasilis Gkatzelis. Approximating the nash social welfare with indivisible items. In STOC, pages 371-380. ACM, 2015. Google Scholar
  9. Shmuel Friedland and Leonid Gurvits. Lower bounds for partial matchings in regular bipartite graphs and applications to the monomer-dimer entropy. Combinatorics, Probability and Computing, 17(03):347-361, 2008. Google Scholar
  10. Osman Güler. Hyperbolic polynomials and interior point methods for convex programming. MOR, 22(2):350-77, 1997. Google Scholar
  11. Leonid Gurvits. Hyperbolic polynomials approach to van der waerden/schrijver-valiant like conjectures: Sharper bounds, simpler proofs and algorithmic applications. In STOC, STOC '06, pages 417-426, 2006. Google Scholar
  12. Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM, 51(4):671-697, July 2004. Google Scholar
  13. Euiwoong Lee. Apx-hardness of maximizing nash social welfare with indivisible items. arXiv preprint arXiv:1507.01159, 2015. Google Scholar
  14. Hervé Moulin. Fair division and collective welfare. MIT press, 2004. Google Scholar
  15. Nhan-Tam Nguyen, Trung Thanh Nguyen, Magnus Roos, and Jörg Rothe. Computational complexity and approximability of social welfare optimization in multiagent resource allocation. Autonomous agents and multi-agent systems, 28(2):256-289, 2014. Google Scholar
  16. Trung Thanh Nguyen and Jörg Rothe. Minimizing envy and maximizing average nash social welfare in the allocation of indivisible goods. Discrete Applied Mathematics, 179:54-68, 2014. Google Scholar
  17. Aleksandar Nikolov and Mohit Singh. Maximizing determinants under partition constraints. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 192-201, 2016. Google Scholar
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