The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions. However, when such applications involve sensitive data about individuals, their privacy concerns should be addressed. In this paper, we study the problem of maximizing monotone submodular functions subject to matroid constraints in the framework of differential privacy. We provide $(1-\frac{1}{\mathrm{e}})$-approximation algorithm which improves upon the previous results in terms of approximation guarantee. This is done with an almost cubic number of function evaluations in our algorithm. Moreover, we study $k$-submodularity, a natural generalization of submodularity. We give the first $\frac{1}{2}$-approximation algorithm that preserves differential privacy for maximizing monotone $k$-submodular functions subject to matroid constraints. The approximation ratio is asymptotically tight and is obtained with an almost linear number of function evaluations.

中文翻译：

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具有拟阵约束的快速私有子模块和 $k$-子模块函数最大化

近十年来，人们对在一定约束下最大化非负单调子模函数的问题进行了深入研究，并针对该问题开发了广泛的有效逼近算法。许多机器学习问题，包括数据汇总和影响最大化，可以自然地建模为最大化单调子模函数的问题。但是，当此类应用程序涉及有关个人的敏感数据时，应解决他们的隐私问题。在本文中，我们研究了在差分隐私框架下最大化受拟阵约束的单调子模函数的问题。我们提供了 $(1-\frac{1}{\mathrm{e}})$-近似算法，它在近似保证方面改进了之前的结果。这是通过我们算法中几乎立方数的函数评估来完成的。此外，我们研究了 $k$-submodularity，这是 submodularity 的自然概括。我们给出了第一个 $\frac{1}{2}$-近似算法，该算法保留了差分隐私，以最大化受拟阵约束的单调 $k$-submodular 函数。近似比率是渐近紧密的，并且是通过函数评估的几乎线性数量获得的。