In the Bidder Selection Problem (BSP) there is a large pool of $n$ potential advertisers competing for ad slots on the user's web page. Due to strict computational restrictions, the advertising platform can run a proper auction only for a fraction $k<n$ of advertisers. We consider the basic optimization problem underlying BSP: given $n$ independent prior distributions, how to efficiently find a subset of $k$ with the objective of either maximizing expected social welfare or revenue of the platform. We study BSP in the classic multi-winner model of position auctions for welfare and revenue objectives using the optimal (respectively, VCG mechanism, or Myerson's auction) format for the selected set of bidders. Previous PTAS results for BSP optimization were only known for single-item auctions and in case of [Segev and Singla 2021] for $l$-unit auctions. More importantly, all of these PTASes were computational complexity results with impractically large running times, which defeats the purpose of using these algorithms under severe computational constraints. We propose a novel Poisson relaxation of BSP for position auctions that immediately implies that 1) BSP is polynomial-time solvable up to a vanishingly small error as the problem size $k$ grows; 2) there is a PTAS for position auctions after combining our relaxation with the trivial brute force algorithm. Unlike all previous PTASes, we implemented our algorithm and did extensive numerical experiments on practically relevant input sizes. First, our experiments corroborate the previous experimental findings of Mehta et al. that a few simple heuristics used in practice perform surprisingly well in terms of approximation factor. Furthermore, our algorithm outperforms Greedy both in running time and approximation on medium and large-sized instances.

翻译：在投标者选择问题（BSP）中，存在一个由$n$个潜在广告商组成的大型池子，他们竞争用户网页上的广告位。由于严格的计算限制，广告平台只能对$k<n$个广告商进行适当的拍卖。我们考虑BSP背后的基本优化问题：给定$n$个独立的先验分布，如何高效地找到一个包含$k$个投标者的子集，目标是最大化预期社会福利或平台收入。我们研究BSP在经典的多赢家位置拍卖模型中，针对福利和收入目标，使用最优格式（分别为VCG机制或Myerson拍卖）对选定投标者集进行拍卖。之前的BSP优化PTAS结果仅适用于单物品拍卖，以及[Segev和Singla 2021]中的$l$单位拍卖。更重要的是，所有这些PTAS都是计算复杂性结果，运行时间不切实际地长，这违背了在严格计算约束下使用这些算法的目的。我们提出了一种新颖的BSP泊松松弛方法，用于位置拍卖，这立即意味着：1) 随着问题规模$k$的增长，BSP可以在多项式时间内以可忽略的误差求解；2) 将我们的松弛与简单的暴力搜索算法结合后，存在一个针对位置拍卖的PTAS。与所有之前的PTAS不同，我们实现了我们的算法，并在实际相关的输入规模上进行了广泛的数值实验。首先，我们的实验证实了Mehta等人之前的实验发现：实践中使用的几种简单启发式算法在近似比方面表现得出奇地好。此外，我们的算法在中型和大型实例上，在运行时间和近似性能方面均优于贪婪算法。