Computing Simple Mechanisms: Lift-and-Round over Marginal Reduced Forms

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 an 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.