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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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Nima Anari
Shayan Oveis Gharan
Amin Saberi
Mohit Singh

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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

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

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