  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.
