  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.
