Mining cohesive subgraphs in attributed graphs is an essential problem in the domain of graph data analysis. The integration of fairness considerations significantly fuels interest in models and algorithms for mining fairness-aware cohesive subgraphs. Notably, the relative fair clique emerges as a robust model, ensuring not only comprehensive attribute coverage but also greater flexibility in distributing attribute vertices. Motivated by the strength of this model, we for the first time pioneer an investigation into the identification of the maximum relative fair clique in large-scale graphs. We introduce a novel concept of colorful support, which serves as the foundation for two innovative graph reduction techniques. These techniques effectively narrow the graph's size by iteratively removing edges that do not belong to relative fair cliques. Furthermore, a series of upper bounds of the maximum relative fair clique size is proposed by incorporating consideration of vertex attributes and colors. The pruning techniques derived from these upper bounds can significantly trim unnecessary search space during the branch-and-bound procedure. Adding to this, we present a heuristic algorithm with a linear time complexity, employing both a degree-based greedy strategy and a colored degree-based greedy strategy to identify a larger relative fair clique. This heuristic algorithm can serve a dual purpose by aiding in branch pruning, thereby enhancing overall search efficiency. Extensive experiments conducted on six real-life datasets demonstrate the efficiency, scalability, and effectiveness of our algorithms.

翻译：在属性图中挖掘凝聚力子图是图数据分析领域的一个基本问题。公平性考量的融入极大地激发了人们对公平感知凝聚力子图模型及算法的研究兴趣。值得注意的是，相对公平团作为一种稳健的模型，不仅确保了属性的全面覆盖，还在属性顶点的分布上提供了更大的灵活性。受该模型优点的启发，我们首次率先探索了在大规模图中识别最大相对公平团的问题。我们引入了一个新颖的概念——彩色支持度，并以此为基础提出了两种创新的图缩减技术。这些技术通过迭代移除不属于相对公平团的边，有效地缩小了图的规模。此外，通过综合考虑顶点属性和颜色，我们提出了一系列关于最大相对公平团大小的上界。基于这些上界的剪枝技术能够在分支定界过程中显著减少不必要的搜索空间。在此基础上，我们提出了一种线性时间复杂度的启发式算法，该算法采用基于度的贪心策略和基于彩色度的贪心策略来识别较大的相对公平团。这种启发式算法可服务于双重目的，即辅助分支剪枝，从而提升整体搜索效率。在六个真实数据集上进行的大量实验证明了我们算法的效率、可扩展性和有效性。