Maximal clique enumeration (MCE) is crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. In this paper, we present a novel approach called HBBMC, a hybrid framework combining vertex-oriented BK branching and edge-oriented BK branching, where the latter adopts a branch-and-bound framework which forms the sub-branches by expanding the partial clique with an edge. This hybrid strategy enables more effective pruning and helps achieve a worst-case time complexity better than the best known one under a condition that holds for the majority of real-world graphs. To further enhance efficiency, we introduce an early termination technique, which leverages the topological information of the graphs and constructs the maximal cliques directly without branching. Our early termination technique is applicable to all branch-and-bound frameworks. Extensive experiments demonstrate the superior performance of our techniques.

翻译：最大团枚举（MCE）对于社区发现和生物网络分析等任务至关重要。现有算法通常采用基于分支定界的框架，并配合以顶点为导向的Bron-Kerbosch（BK）分支策略，该策略通过向部分团中添加一个顶点来形成子分支。本文提出了一种称为HBBMC的新方法，这是一个混合框架，结合了以顶点为导向的BK分支和以边为导向的BK分支，后者采用一种通过向部分团中添加一条边来形成子分支的分支定界框架。这种混合策略能够实现更有效的剪枝，并在一个对大多数现实世界图都成立的条件假设下，帮助获得优于已知最佳的最坏情况时间复杂度。为了进一步提升效率，我们引入了一种提前终止技术，该技术利用图的拓扑信息，无需分支即可直接构建最大团。我们的提前终止技术适用于所有分支定界框架。大量实验证明了我们技术的优越性能。