Extracting cohesive subgraphs from complex networks is a fundamental task in graph analytics and is essential for understanding biological, social, and web graphs. The edge-based γ-quasi-clique model offers a flexible alternative by identifying subgraphs whose edge densities exceed a specified threshold γ. However, finding the exact maximum edge-based quasi-clique is computationally challenging, as the problem is NP-hard and lacks the hereditary property. These characteristics limit the effectiveness of conventional pruning methods and the development of efficient reduction rules. As a result, existing algorithms, such as QClique and FPCE, struggle to scale to large graphs. In this paper, we revisit the problem and propose a novel iterative framework that reformulates the problem as a sequence of hereditary subproblems, enabling more effective pruning and reduction strategies and improving the worst-case time complexity. Furthermore, we redesign the iterative process and introduce a novel heuristic to further improve practical efficiency. Extensive experiments on 253 large-scale real-world graphs demonstrate that our proposed algorithm EQC-Pro outperforms existing methods by up to four orders of magnitude.

Cohesive subgraph mining is a fundamental problem in graph theory with numerous real-world applications, such as social network analysis and protein-protein interaction modeling. Among various cohesive subgraphs, the γ-quasi-clique is widely studied for its flexibility in requiring each vertex to connect to at least a γ proportion of other vertices in the subgraph. However, solving the maximum γ-quasi-clique problem is NP-hard and further complicated by the lack of the hereditary property, which makes designing efficient pruning strategies challenging. Existing algorithms, such as DDA and FastQC, either struggle with scalability or exhibit significant performance declines for small values of γ. In this paper, we propose a novel algorithm, IterQC, which reformulates the maximum γ-quasi-clique problem as a series of k-plex problems that possess the hereditary property. IterQC introduces a non-trivial iterative framework and incorporates two key optimization techniques: (1) the pseudo lower bound (pseudo LB) technique, which leverages information across iterations to improve the efficiency of branch-and-bound searches, and (2) the preprocessing technique that reduces problem size and unnecessary iterations. Extensive experiments demonstrate that IterQC achieves up to four orders of magnitude speedup and solves significantly more graph instances compared to state-of-the-art algorithms DDA and FastQC.