We consider the Max-Buying Problem with Limited Supply, in which there are
n items, with Ci copies of each item i, and m bidders such that every
bidder b has valuation vib for item i. The goal is to find a pricing
p and an allocation of items to bidders that maximizes the profit, where
every item is allocated to at most Ci bidders, every bidder receives at most
one item and if a bidder b receives item i then pi≤vib. Briest
and Krysta presented a 2-approximation for this problem and Aggarwal et al.
presented a 4-approximation for the Price Ladder variant where the pricing must
be non-increasing (that is, p1≥p2≥⋯≥pn). We present an
e/(e−1)-approximation for the Max-Buying Problem with Limited Supply and, for
every ε>0, a (2+ε)-approximation for the Price Ladder
variant