# LP-based Approximation for Personalized Reserve Prices

## TL;DR

This paper introduces a new LP-based approximation method for computing personalized reserve prices in second-price auctions, achieving better approximation ratios than previous algorithms without assuming valuation distributions.

## Contribution

The authors develop a novel LP formulation and rounding technique that improves the approximation factor for personalized reserve prices in auction revenue maximization.

## Key findings

- Achieves approximately 0.684-approximation ratio.
- Improves over the previous 0.5-approximation algorithm.
- Bounds the integrality gap of the LP at 0.828.

## Abstract

We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data-set that contains the submitted bids of $n$ buyers in a set of auctions and the problem is to return personalized reserve prices $\textbf r$ that maximize the revenue earned on these auctions by running eager second price auctions with reserve $\textbf r$. For this problem, which is known to be APX-hard, we present a novel LP formulation and a rounding procedure which achieves a $(1+2(\sqrt{2}-1)e^{\sqrt{2}-2})^{-1} \approx 0.684$-approximation. This improves over the $\frac{1}{2}$-approximation algorithm due to Roughgarden and Wang. We show that our analysis is tight for this rounding procedure. We also bound the integrality gap of the LP, which shows that it is impossible to design an algorithm that yields an approximation factor larger than $0.828$ with respect to this LP.