Approximation Algorithms for the Max-Buying Problem with Limited Supply

Abstract

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 consumers such that each consumer b has a valuation $v_{ib}$ for item i. The goal is to find a pricing p and an allocation of items to consumers that maximize the revenue, with every item allocated to at most $C_i$ consumers, every consumer receives at most one item, and if a consumer b receives item i, then $p_i \le v_{ib}$. We present a randomized $e/(e-1)$-approximation for the Max-Buying Problem with Limited Supply and show how to derandomize it, improving the previously known upper bound of 2. The algorithm uses an integer programming formulation with an exponential number of variables to do a probabilistic rounding and it explores some structure of the problem that might be useful when developing approximations for other pricing problems. We also present a PTAS for the price ladder variant, in which the pricing must be non-increasing (that is, $p_1 \ge p_2 \ge \cdots \ge p_n$), improving the previously known upper bound of 4.