For any $\varepsilon>0$, we give a simple, deterministic $(4+\varepsilon)$-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 $e (ω+ 2 + \varepsilon)$-approximation if the ratio between the largest weight and the average weight is at most $ω$. 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 that is both $1/2$-EFX and a $(8+\varepsilon)$-approximation to the symmetric NSW problem under submodular valuations.

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问题。同时，我们考虑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平均数，并在最大权重与平均权重之比不超过 $\omega$ 的条件下，给出一个 $e (\omega + 2 + \varepsilon)$-近似算法。此外，我们证明 $1/2$-EFX 无嫉妒性可与常数因子近似同时实现。具体地，我们能在多项式时间内找到一个分配，该分配同时满足 $1/2$-EFX 性质，并且对子模估值下的对称纳什社会福利问题达到 $(8+\varepsilon)$-近似。