For any $\eps>0$, we give a simple, deterministic $(4+\eps)$-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. 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 $(\omega + 2 + \eps) \ee$-approximation if the ratio between the largest weight and the average weight is at most $\omega$. We also show that the $\nfrac12$-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time that is both $\nfrac12$-EFX and a $(8+\eps)$-approximation to the symmetric NSW problem under submodular valuations.

翻译：对于任意 $\eps>0$，我们针对次模估值下的纳什社会福利（NSW）问题提出了一种简单、确定性的 $(4+\eps)$-近似算法。我们还考虑了该问题的非对称变体，其目标为最大化智能体估值的加权几何平均，并证明当最大权重与平均权重之比至多为 $\omega$ 时，可获得 $(\omega + 2 + \eps) \ee$-近似解。此外，我们证明了 $\nfrac12$-EFX 无嫉妒性质可与常数因子近似同时达成。更精确地说，我们能在多项式时间内找到一种分配方案，该方案在次模估值下既是 $\nfrac12$-EFX 的，也是对对称 NSW 问题的 $(8+\eps)$-近似解。