Nash Social Welfare, Matrix Permanent, and Stable Polynomials


We study the problem of allocatingm 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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@article{Anari2016NashSW, title={Nash Social Welfare, Matrix Permanent, and Stable Polynomials}, author={Nima Anari and Shayan Oveis Gharan and Amin Saberi and Mohit Singh}, journal={CoRR}, year={2016}, volume={abs/1609.07056} }