Submodular maximization subject to matroid constraints is a central problem with many applications in machine learning. As algorithms are increasingly used in decision-making over datapoints with sensitive attributes such as gender or race, it is becoming crucial to enforce fairness to avoid bias and discrimination. Recent work has addressed the challenge of developing efficient approximation algorithms for fair matroid submodular maximization. However, the best algorithms known so far are only guaranteed to satisfy a relaxed version of the fairness constraints that loses a factor 2, i.e., the problem may ask for $\ell$ elements with a given attribute, but the algorithm is only guaranteed to find $\lfloor \ell/2 \rfloor$. In particular, there is no provable guarantee when $\ell=1$, which corresponds to a key special case of perfect matching constraints. In this work, we achieve a new trade-off via an algorithm that gets arbitrarily close to full fairness. Namely, for any constant $\varepsilon>0$, we give a constant-factor approximation to fair monotone matroid submodular maximization that in expectation loses only a factor $(1-\varepsilon)$ in the lower-bound fairness constraint. Our empirical evaluation on a standard suite of real-world datasets -- including clustering, recommendation, and coverage tasks -- demonstrates the practical effectiveness of our methods.

翻译：在拟阵约束下的子模最大化是机器学习中具有众多应用的核心问题。随着算法在涉及性别或种族等敏感属性的数据点决策中日益广泛应用，确保公平性以避免偏见和歧视变得至关重要。近期研究致力于为公平拟阵子模最大化问题开发高效近似算法。然而，目前已知的最佳算法仅能保证满足松弛版本的公平约束，这会损失2倍因子，即问题可能要求包含$\ell$个具有特定属性的元素，但算法仅能保证找到$\lfloor \ell/2 \rfloor$个。特别地，当$\ell=1$时（对应完美匹配约束的关键特例），现有算法缺乏可证明的保证。本研究通过提出一种可无限逼近完全公平的算法，实现了新的权衡效果。具体而言，对于任意常数$\varepsilon>0$，我们给出公平单调拟阵子模最大化的常数因子近似算法，该算法在期望意义上仅损失下界公平约束的$(1-\varepsilon)$倍因子。我们在标准现实数据集套件（包括聚类、推荐和覆盖任务）上的实证评估验证了所提方法的实际有效性。