  We provide a constructive proof of Border's theorem [Bor91, HR15a] and its
generalization to reduced-form auctions with asymmetric bidders [Bor07, MV10,
CKM13]. Given a reduced form, we identify a subset of Border constraints that
are necessary and sufficient to determine its feasibility. Importantly, the
number of these constraints is linear in the total number of bidder types. In
addition, we provide a characterization result showing that every feasible
reduced form can be induced by an ex-post allocation rule that is a
distribution over ironings of the same total ordering of the union of all
bidders' types.
  We show how to leverage our results for single-item reduced forms to design
auctions with heterogeneous items and asymmetric bidders with valuations that
are additive over items. Appealing to our constructive Border's theorem, we
obtain polynomial-time algorithms for computing the revenue-optimal auction.
Appealing to our characterization of feasible reduced forms, we characterize
feasible multi-item allocation rules.
