We consider a revenue-maximizing seller with $m$ heterogeneous items and a single buyer whose valuation $v$ for the items may exhibit both substitutes (i.e., for some $S, T$, $v(S cup T) < v(S) + v(T)$) and complements (i.e., for some $S, T$, $v(S cup T) > v(S) + v(T)$). We show that the mechanism first proposed by Babaioff et al. [2014] - the better of selling the items separately and bundling them together - guarantees a $Theta(d)$ fraction of the optimal revenue, where $d$ is a measure on the degree of complementarity. Note that this is the first approximately optimal mechanism for a buyer whose valuation exhibits any kind of complementarity, and extends the work of Rubinstein and Weinberg [2015], which proved that the same simple mechanisms achieve a constant factor approximation when buyer valuations are subadditive, the most general class of complement-free valuations. Our proof is enabled by the recent duality framework developed in Cai et al. [2016], which we use to obtain a bound on the optimal revenue in this setting. Our main technical contributions are specialized to handle the intricacies of settings with complements, and include an algorithm for partitioning edges in a hypergraph. Even nailing down the right model and notion of degree of complementarity to obtain meaningful results is of interest, as the natural extensions of previous definitions provably fail.
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