In budget-feasible mechanism design, there is a set of items $U$, each owned by a distinct seller. The seller of item $e$ incurs a private cost $\overline{c}_e$ for supplying her item. A buyer wishes to procure a set of items from the sellers of maximum value, where the value of a set $S\subseteq U$ of items is given by a valuation function $v:2^U\to \mathbb{R}_+$. The buyer has a budget of $B \in \mathbb{R}_+$ for the total payments made to the sellers. We wish to design a mechanism that is truthful, that is, sellers are incentivized to report their true costs, budget-feasible, that is, the sum of the payments made to the sellers is at most the budget $B$, and that outputs a set whose value is large compared to $\text{OPT}:=\max\{v(S):\overline{c}(S)\le B,S\subseteq U\}$. Budget-feasible mechanism design has been extensively studied, with the literature focussing on (classes of) subadditive valuation functions, and various polytime, budget-feasible mechanisms, achieving constant-factor approximation, have been devised for the special cases of additive, submodular, and XOS valuations. However, for general subadditive valuations, the best-known approximation factor achievable by a polytime budget-feasible mechanism (given access to demand oracles) was only $O(\log n / \log \log n)$, where $n$ is the number of items. We improve this state-of-the-art significantly by designing a randomized budget-feasible mechanism for subadditive valuations that achieves a substantially-improved approximation factor of $O(\log\log n)$ and runs in polynomial time, given access to demand oracles.

翻译：在预算可行机制设计中，存在一个物品集合$U$，其中每个物品由不同的卖家拥有。物品$e$的卖家在提供其物品时需承担一个私有成本$\overline{c}_e$。买家希望从卖家处采购一个价值最大的物品集合，其中物品集合$S\subseteq U$的价值由估值函数$v:2^U\to \mathbb{R}_+$给出。买家用于向卖家支付的总预算为$B \in \mathbb{R}_+$。我们希望设计一种机制，该机制是真实的（即激励卖家如实报告其成本），预算可行的（即向卖家支付的总金额不超过预算$B$），并且能输出一个价值与$\text{OPT}:=\max\{v(S):\overline{c}(S)\le B,S\subseteq U\}$相比足够大的集合。预算可行机制设计已被广泛研究，文献主要关注（各类）次可加估值函数，并且针对加法估值、子模估值和XOS估值等特殊情况，已经设计出多种实现常数因子近似的多项式时间预算可行机制。然而，对于一般的次可加估值，已知的由多项式时间预算可行机制（在给定需求预言机访问权限下）可实现的最佳近似因子仅为$O(\log n / \log \log n)$，其中$n$为物品数量。我们通过为次可加估值设计一种随机化预算可行机制，显著改进了这一最新技术水平，该机制实现了$O(\log\log n)$的近似因子（这是一个实质性改进），并且在给定需求预言机访问权限下可在多项式时间内运行。