Nash Social Welfare, Matrix Permanent, and Stable Polynomials

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.

Joint work with Nima Anari, Shayan Oveis Gharan, and Mohit Singh.