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.

翻译：子模最大化问题（SMK）是离散优化中的基本问题，在机器学习及相关领域具有广泛应用。随着这些应用越来越多地涉及敏感个体数据，对能够提供形式化隐私保障的高效算法需求日益增长。本文研究差分隐私下的子模最大化问题，同时考虑单调与非单调目标函数。针对单调目标函数，我们提出一种差分隐私算法，该算法在显著改进先前工作中加性误差与查询复杂度的同时，实现了最优的$(1-1/e)$-近似比。我们还针对相同设定提出一种更高效的算法，其近似比为$1/2$。针对非单调目标函数，据我们所知，本文首次提出具有可证隐私保障的差分隐私算法，该算法在期望意义下达到$1/4$-近似比，且加性误差与当前单调目标函数最优算法的已知结果相当。