A longstanding open problem in Algorithmic Mechanism Design is to design computationally efficient truthful mechanisms for (approximately) maximizing welfare in combinatorial auctions
with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and
Schapira [STOC’06] who gave an O(log^2m)-approximation where m is the number of items.
This problem has been studied extensively since, culminating in an O(√log m)-approximation
mechanism by Dobzinski [STOC’16].
We present a computationally-efficient truthful mechanism with approximation ratio that
improves upon the state-of-the-art by an exponential factor. In particular, our mechanism
achieves an O((log log m)^3)-approximation in expectation, uses only O(n) demand queries, and
has universal truthfulness guarantee.
This settles an open question of Dobzinski on whether
Θ(√log m) is the best approximation ratio in this setting in negative.