# Nash Social Welfare, Matrix Permanent, and Stable Polynomials

arXiv: https://arxiv.org/abs/1609.07056 · HTML: https://mlsanity.com/paper/1609.07056/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2016.

## Built on by (most cited first, 1 of 1)

- [Price of Fairness in Bandits: A Tight Minimax Characterization](https://mlsanity.com/paper/2607.13402/) (2026)

## Abstract (arXiv)

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.