  Consider a monopolist selling $n$ items to an additive buyer whose item
values are drawn from independent distributions $F_1,F_2,\ldots,F_n$ possibly
having unbounded support. Unlike in the single-item case, it is well known that
the revenue-optimal selling mechanism (a pricing scheme) may be complex,
sometimes requiring a continuum of menu entries. Also known is that simple
mechanisms with a bounded number of menu entries can extract a constant
fraction of the optimal revenue. Nonetheless, whether an arbitrarily high
fraction of the optimal revenue can be extracted via a bounded menu size
remained open.
  We give an affirmative answer: for every $n$ and $\varepsilon>0$, there
exists $C=C(n,\varepsilon)$ s.t. mechanisms of menu size at most $C$ suffice
for obtaining $(1-\varepsilon)$ of the optimal revenue from any
$F_1,\ldots,F_n$. We prove upper and lower bounds on the revenue-approximation
complexity $C(n,\varepsilon)$ and on the deterministic communication complexity
required to run a mechanism achieving such an approximation.
