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We study revenue maximization in multi-item multi-bidder auctions under thenatural item-independence assumption - a classical problem in Multi-DimensionalBayesian Mechanism Design. One of the biggest challenges in this area isdeveloping algorithms to compute (approximately) optimal mechanisms that arenot brute-force in the size of the bidder type space, which is usuallyexponential in the number of items in multi-item auctions. Unfortunately, suchalgorithms were only known for basic settings of our problem when bidders haveunit-demand [CHMS10,CMS15] or additive valuations [Yao15].  In this paper, we significantly improve the previous results and design thefirst algorithm that runs in time polynomial in the number of items and thenumber of bidders to compute mechanisms that are $O(1)$-approximations to theoptimal revenue when bidders have XOS valuations, resolving an open problemraised 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 acorollary of our result, we show how to compute an approximately optimal andsimple mechanism efficiently using only sample access to the bidders' valuedistributions. Our algorithm builds on two innovations that allow us to searchover the space of mechanisms efficiently: (i) a new type of succinctrepresentation of mechanisms - the marginal reduced forms, and (ii) a novelLift-and-Round procedure that concavifies the problem.

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