  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.
