Complements between goods - where one good takes on added value in the
presence of another - have been a thorn in the side of algorithmic mechanism
designers. On the one hand, complements are common in the standard motivating
applications for combinatorial auctions, like spectrum license auctions. On the
other, welfare maximization in the presence of complements is notoriously
difficult, and this intractability has stymied theoretical progress in the
area.
We define a new interactive differentially private mechanism -- the median
mechanism -- for answering arbitrary predicate queries that arrive online.
Relative to fixed accuracy and privacy constraints, this mechanism can answer
exponentially more queries than the previously best known interactive privacy
mechanism (the Laplace mechanism, which independently perturbs each query
result). Our guarantee is almost the best possible, even for non-interactive
privacy mechanisms.