We present a computationally-efficient truthful mechanism for combinatorial auctions with subadditive bidders that achieves an O((log log m)^3)-approximation to the maximum welfare in expectation
using O(n) demand queries; here m and n are the number of items and bidders, respectively. This breaks the longstanding logarithmic barrier for the problem dating back
to the O(log m loglog m)-approximation mechanism of Dobzinski from 2007. Along the way, we also improve and considerably simplify the state-of-the-art mechanisms for submodular bidders.