We analyze the computational and communication complexity of combinatorial auctions from a new perspective: the degree of interdependency between the items for sale in the bidders’ preferences. Denoting by Gk the class of valuations displaying up to k-wise dependencies, we consider the hierarchy G1 < G2 < ... < Gm, where m is the number of items for sale. We show that the minimum non-trivial degree of interdependency (2-wise dependency) is sufficient to render NP-hard the problem of computing the optimal allocation (but we also exhibit a restricted class of such valuations for which computing the optimal allocation is easy). On the other hand, bidders’ preferences can be communicated efficiently (i.e., exchanging a polynomial amount of information) as long as the interdependencies between items are limited to sets of cardinality up to k, where k is an arbitrary constant. The amount of communication required to transmit the bidders’ preferences becomes super-polynomial (under the assumption that only value queries are allowed) when interdependencies occur between sets of cardinality g(m), where g(m) is an arbitrary function such that g(m) goes to infinity as m goes to infinity. We also consider approximate elicitation, in which the auctioneer learns, asking polynomially many value queries, an approximation of the bidders’ actual preferences.