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arXiv · 1609.07056

Nash Social Welfare, Matrix Permanent, and Stable Polynomials

Nima Anari·Shayan Oveis Gharan·Amin Saberi·Mohit Singh
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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. 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, Mohit Singh. 2016-09-22. Nash Social Welfare, Matrix Permanent, and Stable Polynomials. https://arxiv.org/abs/1609.07056

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