Differentially Private Submodular Maximization with a Knapsack Constraint

2026-06-12

AI Summary

This work studies the problem of maximizing submodular functions subject to a knapsack constraint under differential privacy, covering both monotone and non-monotone settings. The authors propose efficient approximation algorithms that achieve an optimal $(1-1/e)$-approximation ratio in the monotone case while significantly reducing additive error and query complexity. For the non-monotone setting, they present the first differentially private algorithm with theoretical guarantees, attaining an expected approximation ratio of $1/4$ and additive error comparable to the best-known monotone algorithms. Their approach integrates differential privacy mechanisms with submodular optimization techniques, knapsack constraint handling, and randomized sampling combined with noise injection.

Abstract

Submodular maximization subject to a knapsack constraint (SMK) is a fundamental problem in discrete optimization, with wide-ranging applications in machine learning and related fields. As these applications increasingly involve sensitive individual data, there is a growing need for high-utility algorithms that provide formal privacy guarantees. In this work, we study the SMK problem under differential privacy, considering both monotone and non-monotone objective functions. For monotone objectives, we propose a differentially private algorithm that achieves the optimal $(1-1/e)$-approximation ratio while significantly improving both additive error and query complexity over prior work. We also present a more efficient algorithm for the same setting, achieving a $1/2$-approximation. For non-monotone objectives, we introduce, to our knowledge, the first differentially private algorithm with provable guarantees, achieving a $1/4$-approximation in expectation and an additive error comparable to the best known for monotone objective functions.

Problem

• Differential Privacy
• Submodular Maximization
• Knapsack Constraint
• Monotone Function
• Non-monotone Function

Innovation

• Differentially Private
• Submodular Maximization
• Knapsack Constraint
• Approximation Algorithm
• Non-monotone Functions