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设置下，受群体平等约束的经典子模最大化问题。已有研究表明，许多机器学习应用（包括数据摘要、社交网络影响最大化及个性化推荐）中的效用函数均满足子模性。因此，在多种约束条件下最大化子模函数成为这些应用的核心问题。从宏观角度看，子模最大化旨在选取最具代表性的项目组（如数据点）。然而，现有算法框架大多未纳入公平性约束，导致特定群体的代表性不足或过度。这促使我们研究带群体平等约束的子模最大化问题——旨在选取项目组以最大化（可能非单调的）子模效用函数的同时满足群体平等约束。为此，我们提出了首个针对该问题的常数因子近似算法。该算法设计具有高度鲁棒性，可扩展至更复杂的自适应环境下的子模最大化问题。此外，我们进一步将研究扩展到全局基数约束及其他公平性指标。