  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.
