Normalized-cut graph partitioning aims to divide the set of nodes in a graph into $k$ disjoint clusters to minimize the fraction of the total edges between any cluster and all other clusters. In this paper, we consider a fair variant of the partitioning problem wherein nodes are characterized by a categorical sensitive attribute (e.g., gender or race) indicating membership to different demographic groups. Our goal is to ensure that each group is approximately proportionally represented in each cluster while minimizing the normalized cut value. To resolve this problem, we propose a two-phase spectral algorithm called FNM. In the first phase, we add an augmented Lagrangian term based on our fairness criteria to the objective function for obtaining a fairer spectral node embedding. Then, in the second phase, we design a rounding scheme to produce $k$ clusters from the fair embedding that effectively trades off fairness and partition quality. Through comprehensive experiments on nine benchmark datasets, we demonstrate the superior performance of FNM compared with three baseline methods.

翻译：[摘要] 归一化割图划分旨在将图中的节点集合划分为\(k\)个互不相交的簇，以最小化任意簇与其他所有簇之间总边数的比例。本文考虑该划分问题的一个公平变体，其中节点由分类敏感属性（如性别或种族）刻画，指示其属于不同人口统计群体。我们的目标是确保每个群体在每个簇中近似按比例代表，同时最小化归一化割值。为解决该问题，我们提出一种名为FNM的两阶段谱算法：第一阶段，在目标函数中加入基于公平性准则的增广拉格朗日项，以获得更公平的谱节点嵌入；第二阶段，设计一种舍入方案，从公平嵌入中生成\(k\)个簇，有效权衡公平性与划分质量。通过在九个基准数据集上的全面实验，我们证明FNM相较于三种基线方法具有优越性能。