Approximation Algorithms for the Max-Buying Problem with Limited Supply

TL;DR

This paper introduces improved approximation algorithms for the Max-Buying Problem with Limited Supply and its Price Ladder variant, achieving better approximation ratios than previous methods.

Contribution

It presents an $e/(e-1)$-approximation for the Max-Buying Problem with Limited Supply and a $(2+ ext{epsilon})$-approximation for the Price Ladder variant, surpassing prior results.

Findings

Achieved an $e/(e-1)$-approximation for the main problem.

Developed a $(2+ ext{epsilon})$-approximation for the Price Ladder variant.

Improved upon previous 2- and 4-approximation algorithms.

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 $m$ 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 maximizes 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 $p_i \leq v_{ib}$. 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, $p_1 \geq p_2 \geq \cdots \geq p_n$). We present an $e/(e-1)$-approximation for the Max-Buying Problem with Limited Supply and, for every $\varepsilon > 0$, a $(2+\varepsilon)$-approximation for the Price Ladder variant.