We consider the Max-Buying Problem with Limited Supply, in which there are n items, with C_i copies of each item i, and m bidders such that every bidder b has valuation v_(ib) for item i. The goal is to find a pricing p and an allocation of items to bidders that maximize the profit, where every item is allocated to at most C_i bidders, every bidder receives at most one item and if a bidder b receives item i then p_i ≤ v_(ib). 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, p_1 ≥ p_2 ≥ … ≥ p_n). We present a randomized 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.
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