We consider a revenue-maximizing single seller with $m$ items for sale to a single buyer whose value $v(\cdot)$ for the items is drawn from a known distribution $D$ of support $k$. A series of works by Cai et al. establishes that when each $v(\cdot)$ in the support of $D$ is additive or unit-demand (or $c$-demand), the revenue-optimal auction can be found in $\operatorname{poly}(m,k)$ time. We show that going barely beyond this, even to matroid-based valuations (a proper subset of Gross Substitutes), results in strong hardness of approximation. Specifically, even on instances with $m$ items and $k \leq m$ valuations in the support of $D$, it is not possible to achieve a $1/m^{1-\varepsilon}$-approximation for any $\varepsilon>0$ to the revenue-optimal mechanism for matroid-based valuations in (randomized) poly-time unless NP $\subseteq$ RP (note that a $1/k$-approximation is trivial). Cai et al.'s main technical contribution is a black-box reduction from revenue maximization for valuations in class $\mathcal{V}$ to optimizing the difference between two values in class $\mathcal{V}$. Our main technical contribution is a black-box reduction in the other direction (for a wide class of valuation classes), establishing that their reduction is essentially tight.

中文翻译：

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关于组合买方贝叶斯收益最大化的（in）-近似性

我们考虑一个收入最大化的单一卖家，将 $m$ 的商品出售给单个买家，该买家的商品价值 $v(\cdot)$ 来自支持 $k$ 的已知分布 $D$。蔡等人的一系列作品。确定当支持 $D$ 的每个 $v(\cdot)$ 是可加的或单位需求（或 $c$-需求）时，收入最优拍卖可以在 $\operatorname{poly}(m ,k)$ 时间。我们表明，即使是基于拟阵的估值（Gross Substitutes 的适当子集）也几乎不超出此范围，会导致很强的近似难度。具体来说，即使在具有 $m$ 项和 $k \leq m$ 估值支持 $D$ 的实例上，也不可能实现 $1/m^{1-\varepsilon}$-对任何 $\伐雷普西隆> 0$ 到（随机）多时间中基于拟阵的估值的收入最优机制，除非 NP $\subseteq$ RP（注意 $1/k$ 近似值是微不足道的）。Cai 等人的主要技术贡献是从 $\mathcal{V}$ 类估值的收入最大化到优化 $\mathcal{V}$ 类中两个值之间的差异的黑盒减少。我们的主要技术贡献是另一个方向的黑箱减少（对于广泛的估值类别），确定它们的减少基本上是紧的。