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.
翻译：我们研究了纳什社会福利问题:考虑到具有估值功能的单位$0:v_i:2_ ⁇ [m]\rightrow ~mathbb R}$2_right $,将 $[m] 分割成$S_1,\ldots,S_n$,以便最大限度地增加$(\ prod ⁇ i=1 ⁇ n} v_i(S_i))\_n}v_i(S_i)}}}。问题已经证明承认添加值、预算增加值和线性线性直线性组合值的常数近似值;次级模式值的情况尚未解决。我们为几种亚模式值估值提供了 $\frac{1}(1\\\\\\\{e}(1\\\\\\\{e})$2$-对以下几类亚模式值的折合值:覆盖面、配料级函数和某些匹配值。
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