Fair top-$k$ selection, which ensures appropriate proportional representation of members from minority or historically disadvantaged groups among the top-$k$ selected candidates, has drawn significant attention. We study the problem of finding a fair (linear) scoring function with multiple protected groups while also minimizing the disparity from a reference scoring function. This generalizes the prior setup, which was restricted to the single-group setting without disparity minimization. Previous studies imply that the number of protected groups may have a limited impact on the runtime efficiency. However, driven by the need for experimental exploration, we find that this implication overlooks a critical issue that may affect the fairness of the outcome. Once this issue is properly considered, our hardness analysis shows that the problem may become computationally intractable even for a two-dimensional dataset and small values of $k$. However, our analysis also reveals a gap in the hardness barrier, enabling us to recover the efficiency for the case of small $k$ when the number of protected groups is sufficiently small. Furthermore, beyond measuring disparity as the "distance" between the fair and the reference scoring functions, we introduce an alternative disparity measure$\unicode{x2014}$utility loss$\unicode{x2014}$that may yield a more stable scoring function under small weight perturbations. Through careful engineering trade-offs that balance implementation complexity, robustness, and performance, our augmented two-pronged solution demonstrates strong empirical performance on real-world datasets, with experimental observations also informing algorithm design and implementation decisions.

翻译：公平Top-$k$选择旨在确保少数群体或历史上处于不利地位的群体成员在Top-$k$入选者中获得恰当的比例代表，这一问题已引起广泛关注。我们研究了在存在多个受保护群体的场景下，寻找一个公平（线性）评分函数，同时最小化其与参考评分函数之间差异的问题。这推广了先前局限于单群体设定且未考虑差异最小化的研究框架。已有研究表明受保护群体数量对运行效率的影响可能有限，但基于实验探索的需求，我们发现这一推论忽视了一个可能影响结果公平性的关键问题。一旦充分考虑该问题，我们的难度分析表明，即使对于二维数据集和较小的$k$值，该问题也可能在计算上变得难以处理。然而，分析同时揭示了难度壁垒中存在间隙，使得当受保护群体数量足够少时，我们能够在小$k$值情况下恢复计算效率。此外，除了将差异衡量为公平评分函数与参考评分函数之间的"距离"，我们引入了另一种差异度量——效用损失——该度量可能在微小权重扰动下产生更稳定的评分函数。通过权衡实现复杂度、鲁棒性与性能的精细工程考量，我们提出的增强型双轨解决方案在真实数据集上展现出强大的实证性能，实验观测结果也为算法设计与实现决策提供了参考。