# Near Optimal Jointly Private Packing Algorithms via Dual Multiplicative Weight Update

## TL;DR

This paper introduces a near-optimal, differentially private packing algorithm that improves resource supply requirements and provides theoretical guarantees, along with a linear-time, truthful, online variant.

## Contribution

It presents an improved $(eta, eta)$-jointly differentially private packing algorithm with better resource bounds and a matching hardness result, plus a fast, truthful, online approach.

## Key findings

- Improves resource supply bounds from $ ilde{O}(m^2 / eta au)$ to $ ilde{O}(rac{
oot{2} ext{m}}{eta au})$
- Provides a near-matching lower bound on resource requirements for private algorithms
- Introduces a linear-time, truthful, online private packing algorithm

## Abstract

We present an improved $(\epsilon, \delta)$-jointly differentially private algorithm for packing problems. Our algorithm gives a feasible output that is approximately optimal up to an $\alpha n$ additive factor as long as the supply of each resource is at least $\tilde{O}(\sqrt{m} / \alpha \epsilon)$, where $m$ is the number of resources. This improves the previous result by Hsu et al.~(SODA '16), which requires the total supply to be at least $\tilde{O}(m^2 / \alpha \epsilon)$, and only guarantees approximate feasibility in terms of total violation. Further, we complement our algorithm with an almost matching hardness result, showing that $\Omega(\sqrt{m \ln(1/\delta)} / \alpha \epsilon)$ supply is necessary for any $(\epsilon, \delta)$-jointly differentially private algorithm to compute an approximately optimal packing solution. Finally, we introduce an alternative approach that runs in linear time, is exactly truthful, can be implemented online, and can be $\epsilon$-jointly differentially private, but requires a larger supply of each resource.