Approximation Algorithms for the Weighted Nash Social Welfare via Convex and Non-Convex Programs

In an instance of the weighted Nash Social Welfare problem, we are given a set of m indivisible items, 𝒢, and n agents, 𝒜, where each agent i ∈𝒜 has a valuation v_ij≥ 0 for each item j∈𝒢. In addition, every agent i has a non-negative weight w_i such that the weights collectively sum up to 1. The goal is to find an assignment σ:𝒢→𝒜 that maximizes ∏_i∈𝒜(∑_j∈σ^-1(i) v_ij)^w_i, the product of the weighted valuations of the players. When all the weights equal 1/n, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present a 5·exp(2· D_KL(𝐰 || 1⃗/n)) = 5·exp(2logn + 2∑_i=1^n w_i logw_i)-approximation algorithm for the weighted Nash Social Welfare problem, where D_KL(𝐰 || 1⃗/n) denotes the KL-divergence between the distribution induced by 𝐰 and the uniform distribution on [n]. We show a novel connection between the convex programming relaxations for the unweighted variant of Nash Social Welfare presented in , and generalize the programs to two different mathematical programs for the weighted case. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation.