nash social welfare matrix permanent and stable polynomials

• DOI: 10.4230/lipics.itcs.2017.36
• Publication Date: 2016-01-01
• Authors: Anari, Nima; Oveis Gharan, Shayan; Saberi, Amin; Singh, Mohit
• Keywords: FOS: Computer and information sciences; Discrete Mathematics (cs.DM); 1. No poverty; Saddle Point; 02 engineering and technology; 004; Computer Science - Data Structures and Algorithms; Randomized Algorithm; 0202 electrical engineering, electronic engineering, information engineering; FOS: Mathematics; Permanent; Matching; Mathematics - Combinatorics; Nash Welfare; Data Structures and Algorithms (cs.DS); Combinatorics (math.CO); ddc:004; 10. No inequality; Stable Polynomial; Computer Science - Discrete Mathematics

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.