For any $\varepsilon>0$, we give a simple, deterministic $(4+\varepsilon)$-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 $(\omega + 2 +\varepsilon) e$-approximation if the ratio between the largest weight and the average weight is at most $\omega$. We also show that the $1/2$-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time which is both $1/2$-EFX and a $(8+\varepsilon)$-approximation to the symmetric NSW problem under submodular valuations. The previous best approximation factor under $1/2$-EFX was linear in the number of agents.

翻译：对于任意 $\varepsilon>0$，我们给出一个简单、确定性的 $(4+\varepsilon)$-近似算法，用于解决子模估值下的纳什社会福利（NSW）问题。此前的最佳近似因子为 $380$，对应一个随机算法。我们还考虑了该问题的非对称变体，其目标是最小化代理人估值的加权几何均值，并给出一个 $(\omega + 2 +\varepsilon) e$-近似算法，其中最大权重与平均权重之比不超过 $\omega$。我们进一步证明，$1/2$-EFX 无嫉妒性质可以与常数因子近似同时实现。更准确地说，我们可以在多项式时间内找到一个既满足 $1/2$-EFX 又达到 $(8+\varepsilon)$-近似于子模估值下对称 NSW 问题的分配方案。此前在 $1/2$-EFX 条件下的最佳近似因子与代理人数量呈线性关系。