We consider Bidder Selection Problem (BSP) in position auctions motivated by practical concerns of online advertising platforms. In this problem, the platform sells ad slots via an auction to a large pool of $n$ potential buyers with independent values drawn from known prior distributions. The seller can only invite a fraction of $k<n$ advertisers to the auction due to communication and computation restrictions. She wishes to maximize either the social welfare or her revenue by selecting the set of invited bidders. We study BSP in a 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. We propose a novel Poisson-Chernoff relaxation of the problem that immediately implies that 1) BSP is polynomial time solvable up to a vanishingly small error as the problem size $k$ grows; 2) PTAS for position auctions after combining our relaxation with the trivial brute force algorithm; the algorithm is in fact an Efficient PTAS (EPTAS) under a mild assumption $k\ge\log n$ with much better running time than previous PTASes for single-item auction. Our approach yields simple and practically relevant algorithms unlike all previous complex PTASes, which had at least doubly exponential dependency of their running time on $\varepsilon$. In contrast, our algorithms are even faster than popular algorithms such as greedy for submodular maximization. Furthermore, we did extensive numerical experiments, which demonstrate high efficiency and practical applicability of our solution. Our experiments corroborate the experimental findings of [Mehta, Nadav, Psomas, Rubinstein 2020] that many simple heuristics perform surprisingly well, which indicates importance of using small $\varepsilon$ for the BSP and practical irrelevance of all previous PTAS approaches.

翻译：我们研究位次拍卖中的竞标者选择问题（BSP），该问题源于在线广告平台的实际需求。在此问题中，平台通过拍卖将广告位售予一个包含$n$个潜在买家的庞大集合，这些买家的独立估值取自已知先验分布。由于通信和计算限制，卖家只能邀请$k<n$个广告商参与拍卖。她希望通过选择受邀竞标者的集合来最大化社会福利或自身收益。我们针对位次拍卖这一经典多获胜者模型，在福利和收益目标下，对所选竞标者集合采用最优格式（分别为VCG机制或Myerson拍卖）来研究BSP。我们提出了一种新颖的泊松-切尔诺夫松弛方法，该方法直接表明：1) 随着问题规模$k$增大，BSP可在多项式时间内求解，且误差可忽略不计；2) 将我们的松弛方法与朴素暴力算法结合后，可为位次拍卖提供多项式时间近似方案（PTAS）；在温和假设$k\ge\log n$下，该算法实际上是一个高效多项式时间近似方案（EPTAS），其运行时间远优于先前针对单物品拍卖的PTAS。我们的方法产生了简单且具实际相关性的算法，与此前所有至少具有双指数级对$\varepsilon$依赖运行时间的复杂PTAS形成鲜明对比。相反，我们的算法甚至比贪心算法（用于子模最大化）等流行算法更快。此外，我们进行了大量数值实验，展示了解决方案的高效性和实际适用性。我们的实验证实了[Mehta, Nadav, Psomas, Rubinstein 2020]的实验发现，即许多简单启发式方法表现异常出色，这表明对BSP使用较小的$\varepsilon$至关重要，而所有先前的PTAS方法在实际中均无关紧要。