We study revenue maximization in multi-item multi-bidder auctions under the natural item-independence assumption - a classical problem in Multi-Dimensional Bayesian Mechanism Design. One of the biggest challenges in this area is developing algorithms to compute (approximately) optimal mechanisms that are not brute-force in the size of the bidder type space, which is usually exponential in the number of items in multi-item auctions. Unfortunately, such algorithms were only known for basic settings of our problem when bidders have unit-demand [CHMS10,CMS15] or additive valuations [Yao15]. In this paper, we significantly improve the previous results and design the first algorithm that runs in time polynomial in the number of items and the number of bidders to compute mechanisms that are $O(1)$-approximations to the optimal revenue when bidders have XOS valuations, resolving the open problem raised in [CM16,CZ17]. Moreover, the computed mechanism has a simple structure: It is either a posted price mechanism or a two-part tariff mechanism. As a corollary of our result, we show how to compute an approximately optimal and simple mechanism efficiently using only sample access to the bidders' value distributions. Our algorithm builds on two innovations that allow us to search over the space of mechanisms efficiently: (i) a new type of succinct representation of mechanisms - the marginal reduced forms, and (ii) a novel Lift-and-Round procedure that concavifies the problem.

翻译：在自然项目独立假设下,我们研究在多项目多投标拍卖中实现收入最大化的问题 -- -- 多多功能贝叶赛亚机制设计中的一个典型问题。这一领域的最大挑战之一是制定算法,计算(大约)并非投标人类型空间规模的强力最佳机制,这种算法通常在多项目拍卖中的项目数量上成倍增长。不幸的是,这种算法只有在投标人有单位-需求[CHMS10,CMS15]或添加值[Yao15]时,我们的问题的基本环境才知道。在本文中,我们大大改进了先前的结果,设计了第一个算法,在项目数量和投标人数量上以时间混合计算(约)最佳机制,在投标人有XOS估值时,这种机制与最佳收入相匹配,解决[CM16,CZ17]中出现的公开问题。此外,计算机制有一个简单的结构:要么是贴现的边际价格机制,要么是两个部分的关税机制。作为我们结果的必然结果,我们展示了如何通过两个层次的抽样机制来进行最优化的搜索机制。