Nash Social Welfare with Submodular Valuations: Approximation Algorithms and Integrality Gaps

We study the problem of allocating items to agents with submodular valuations with the goal of maximizing the weighted Nash social welfare (NSW). The best-known results for unweighted and weighted objectives are the $(4+\epsilon)$ approximation given by Garg, Husic, Li, V\'egh, and Vondr\'ak~[STOC 2023] and the $(233+\epsilon)$ approximation given by Feng, Hu, Li, and Zhang~[STOC 2025], respectively. In this work, we present a $(3.56+\epsilon)$-approximation algorithm for weighted NSW maximization with submodular valuations, simultaneously improving the previous approximation ratios of both the weighted and unweighted NSW problems. Our algorithm solves the configuration LP of Feng, Hu, Li, and Zhang~[STOC 2025] via a stronger separation oracle that loses an $e/(e-1)$ factor only on small items, and then rounds the solution via a new bipartite multigraph construction. Some key technical ingredients of our analysis include a greedy proxy function, additive within each configuration, that preserves the LP value while lower-bounding the rounded solution, together with refined concentration bounds and a series of mathematical programs analyzed partly by computer assistance. On the hardness side, we prove that the configuration LP for weighted NSW with submodular valuations has an integrality gap of at least $(2^{\ln 2}-\epsilon) \approx 1.617 - \epsilon$, which is larger than the current best-known $e/(e-1)-\epsilon \approx 1.582-\epsilon$ hardness~[SODA 2020]. For additive valuations, we show an integrality gap of $(e^{1/e}-\epsilon)$, which proves the tightness of the approximation ratio in~[ICALP 2024] for algorithms based on the configuration LP. For unweighted NSW with additive valuations, we show an integrality gap of $(2^{1/4}-\epsilon) \approx 1.189-\epsilon$, again larger than the current best-known $\sqrt{8/7} \approx 1.069$-hardness for the problem~[Math. Oper. Res. 2024].