Approximating Nash Social Welfare by Matching and Local Search

For any >0, we give a simple, deterministic (4+)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. The previous best approximation factor was 380 via a randomized algorithm. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents’ valuations, and give an (ω + 2 + ) -approximation if the ratio between the largest weight and the average weight is at most ω.