In this paper, we study the classic submodular maximization problem subject to a group equality constraint under both non-adaptive and adaptive settings. It has been shown that the utility function of many machine learning applications, including data summarization, influence maximization in social networks, and personalized recommendation, satisfies the property of submodularity. Hence, maximizing a submodular function subject to various constraints can be found at the heart of many of those applications. On a high level, submodular maximization aims to select a group of most representative items (e.g., data points). However, the design of most existing algorithms does not incorporate the fairness constraint, leading to under- or over-representation of some particular groups. This motivates us to study the submodular maximization problem with group equality, where we aim to select a group of items to maximize a (possibly non-monotone) submodular utility function subject to a group equality constraint. To this end, we develop the first constant-factor approximation algorithm for this problem. The design of our algorithm is robust enough to be extended to solving the submodular maximization problem under a more complicated adaptive setting. Moreover, we further extend our study to incorporating a global cardinality constraint and other fairness notations.

翻译：本文研究了在非自适应和自适应两种设置下，受群体公平性约束的经典子模最大化问题。已有研究表明，包括数据摘要、社交网络影响力最大化和个性化推荐在内的许多机器学习应用的效用函数都满足子模性质。因此，在多种约束条件下最大化子模函数构成了这些应用的核心。从高层次来看，子模最大化旨在选择一组最具代表性的项目（如数据点）。然而，现有大多数算法的设计并未纳入公平性约束，导致某些特定群体被过度或不足代表。这促使我们研究带群体公平性的子模最大化问题，其目标是在群体公平性约束下，选择一组项目以最大化（可能非单调的）子模效用函数。为此，我们提出了该问题的首个常数因子近似算法。该算法设计具有足够鲁棒性，可扩展至解决更复杂的自适应设置下的子模最大化问题。此外，我们进一步将研究拓展到包含全局基数约束及其他公平性概念的情形。