AI 中文总结
本文研究组合拍卖中占优策略激励相容机制的通信复杂度，统一证明了一般、XOS和总替代估值下近似比的下界，并解决了总替代情形下的开放问题，同时给出双投标人设置中的新结果。
AI 中文摘要
我们研究了组合拍卖中占优策略激励相容（DSIC）机制的通信复杂度。对于γ∈[log m, m]，令DSIC_GEN(m, γ)、DSIC_XOS(m, γ)和DSIC_GS(m, γ)分别表示在m个物品上使用至多2^γ通信的确定性、个体理性、无负转移的DSIC机制，对于一般单调、XOS和总替代（GS）估值所能达到的最佳近似比。我们给出了一个统一证明，表明DSIC_GEN(m, γ)=Ω(m/γ)，DSIC_XOS(m, γ)=Ω((m/γ)^{1/5})，以及DSIC_GS(m, γ)=Ω((m/γ)^{1/7})。GS下界回答了DobzinskiRV22的一个开放问题：尽管GS估值的多项式通信福利最大化通过VCG机制允许多项式通信的确定性真实机制，但对于确定性DSIC机制，在多项式通信下不可能获得好的近似。此外，一般下界确立了DSIC_GEN(m, γ)=Θ(m/γ)，这得益于QiuW24的确定性DSIC O(m/γ)-近似。我们还在双投标人设置中获得了几项结果。我们表明，对于两个加权拟阵秩估值（GS的一个子类），使用普遍DSIC机制达到1.0001-近似需要指数通信。另一方面，我们给出了一个多项式通信的(1+√5)/2≈1.618-近似，用于两个子模投标人，使用普遍DSIC机制。在此工作之前，尚不清楚对于两个子模估值，多项式通信的普遍真实机制是否能击败2-近似，也不清楚对于两个加权拟阵秩估值，多项式通信的普遍DSIC机制是否能击败2-近似。
英文摘要
We study the communication complexity of \emph{dominant-strategy incentive-compatible} (DSIC) mechanisms for combinatorial auctions. For $γ\in [\log m, m]$, let $\mathsf{DSIC_{GEN}}(m, γ)$, $\mathsf{DSIC_{XOS}}(m, γ)$, and $\mathsf{DSIC_{GS}}(m, γ)$ denote the best approximation ratio attainable by a deterministic, individually rational, no-negative-transfers, DSIC mechanism using at most $2^γ$ communication over $m$ items, for general monotone, XOS, and gross substitutes (GS) valuations, respectively. We give a unified proof that shows $\mathsf{DSIC_{GEN}}(m, γ) = Ω(m/γ)$, $\mathsf{DSIC_{XOS}}(m, γ) = Ω((m/γ)^{1/5})$, and $\mathsf{DSIC_{GS}}(m, γ) = Ω((m/γ)^{1/7})$. The GS lower bound answers an open question of~\cite{DobzinskiRV22}: although poly-communication welfare maximization for GS valuations admits a poly-communication deterministic truthful mechanism via VCG, no good approximation is possible in poly-communication for deterministic DSIC mechanisms. Additionally, the general lower bound establishes that $\mathsf{DSIC_{GEN}}(m, γ) = Θ(m/γ)$ due to the deterministic DSIC $O(m/γ)$-approximation of~\cite{QiuW24}. We also obtain several results in the two-bidder setting. We show that attaining a $1.0001$-approximation for two weighted matroid-rank valuations (a subclass of GS) with a universally DSIC mechanism requires exponential communication. On the other hand, we give a poly-communication $(1+\sqrt{5})/2 \approx 1.618$-approximation for two submodular bidders using a universally DSIC mechanism. Prior to this work, it was not known whether even a poly-communication universally truthful mechanism could beat a $2$-approximation for two submodular valuations, nor whether a poly-communication universally DSIC mechanism could beat a $2$-approximation for two weighted matroid rank valuations.
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