We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data-set that contains the submitted bids of $n$ buyers in a set of auctions and the problem is to return personalized reserve prices $\textbf r$ that maximize the revenue earned on these auctions by running eager second price auctions with reserve $\textbf r$. For this problem, which is known to be APX-hard, we present a novel LP formulation and a rounding procedure which achieves a $(1+2(\sqrt{2}-1)e^{\sqrt{2}-2})^{-1} \approx 0.684$-approximation. This improves over the $\frac{1}{2}$-approximation algorithm due to Roughgarden and Wang. We show that our analysis is tight for this rounding procedure. We also bound the integrality gap of the LP, which shows that it is impossible to design an algorithm that yields an approximation factor larger than $0.828$ with respect to this LP.

中文翻译：

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基于 LP 的个性化底价近似

我们研究了在急切的第二价格拍卖中计算数据驱动的个性化底价的问题，而无需对估值分布进行任何假设。在这里，输入是一个数据集，其中包含一组拍卖中 $n$ 买家提交的出价，问题是返回个性化的底价 $\textbf r$，通过运行急切的秒来最大化在这些拍卖中获得的收入保留 $\textbf r$ 的价格拍卖。对于这个已知是 APX 难的问题，我们提出了一个新的 LP 公式和一个舍入过程，它实现了 $(1+2(\sqrt{2}-1)e^{\sqrt{2}-2 })^{-1} \approx 0.684$-近似值。由于 Roughgarden 和 Wang，这改进了 $\frac{1}{2}$-approximation 算法。我们表明我们的分析对于这个舍入过程是严格的。我们还限制了 LP 的完整性差距，