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.
@article{arxiv.1609.07056,
title = {Nash Social Welfare, Matrix Permanent, and Stable Polynomials},
author = {Nima Anari and Shayan Oveis Gharan and Amin Saberi and Mohit Singh},
journal= {arXiv preprint arXiv:1609.07056},
year = {2016}
}