The design of truthful auctions that approximate the optimal expected revenue is a central problem in algorithmic mechanism design. 30 years after Myerson's characterization of Bayesian optimal auctions in single-parameter domains [8], characterizing but also providing efficient mechanisms for multi-parameter domains still remains a very important unsolved problem. Our work improves upon recent results in this area, introducing new techniques for tackling the problem, while also combining and extending recently introduced tools. In particular we give the first approximation algorithms for Bayesian auctions with multiple heterogeneous items when bidders have additive valuations, budget constraints and general matroid feasibility constraints.

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