  We study the Nash Social Welfare problem: Given $n$ agents with valuation
functions $v_i:2^{[m]} \rightarrow {\mathbb R}$, partition $[m]$ into
$S_1,\ldots,S_n$ so as to maximize $(\prod_{i=1}^{n} v_i(S_i))^{1/n}$. The
problem has been shown to admit a constant-factor approximation for additive,
budget-additive, and piecewise linear concave separable valuations; the case of
submodular valuations is open.
  We provide a $\frac{1}{e} (1-\frac{1}{e})^2$-approximation of the {\em
optimal value} for several classes of submodular valuations: coverage, sums of
matroid rank functions, and certain matching-based valuations.
