Estimating the Nash Social Welfare for coverage and other submodular valuations

We study the Nash Social Welfare problem: Given $n$ agents with valuation functions $vi:2^{[m]} \rightarrow {\mathbb R}$, partition $[m]$ into $S1,\ldots,Sn$ so as to maximize $(\prod{i=1}^{n} vi(Si))^{1/n}$. 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.