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 . When all the weights equal to , the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present a -approximation algorithm for the weighted Nash Social Welfare problem, where denotes the KL-divergence between the distribution w and the uniform distribution on [n].

We generalize the convex programming relaxations for the symmetric variant of Nash Social Welfare presented in [CDG⁺17, AGSS17] to two different mathematical programs. 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.