The $k$-defective clique model relaxes the strict completeness constraint of the traditional clique by allowing up to $k$ missing edges, providing a robust formulation for detecting cohesive structures in noisy graphs. Consequently, the maximum $k$-defective clique problem has attracted significant attention. State-of-the-art exact algorithms predominantly adopt the branch-and-bound framework, which recursively partitions the current problem instance (or branch) into two sub-problems via a branching procedure, until each sub-problem becomes trivially solvable. However, this strategy often leads to excessive branching by overlooking intermediate sub-problems that are non-trivial yet efficiently solvable. While recent studies have attempted to refine branching procedures, they fail to address this structural redundancy. To address this, we propose BBRes, a framework that incorporates a novel early termination strategy into the recursive branching process. By employing a specialized polynomial-time solver to identify and resolve tractable sub-instances, BBRes effectively avoids redundant branching steps. Additionally, we design a tailored branching strategy that synergizes with this termination mechanism. As a result, BBRes achieves an improved theoretical worst-case time complexity. To enhance practical performance, we propose a tighter upper bound based on a novel double graph coloring method integrated with max-flow techniques, which is orthogonal to the branching framework. Extensive experiments show that BBRes achieves at least 2X speedup over state-of-the-art methods on a substantial fraction of the datasets.

翻译：$k$-缺陷团模型通过允许最多$k$条缺失边，放松了传统团的严格完全性约束，为检测含噪图中的凝聚结构提供了鲁棒性建模框架。因此，最大$k$-缺陷团问题引起了广泛关注。当前最先进的精确算法主要采用分支定界框架，通过递归分支过程将当前问题实例（或分支）划分为两个子问题，直至每个子问题可平凡求解。然而，这种策略常因忽略非平凡但可高效求解的中间子问题而导致过度分支。尽管近期研究尝试改进分支过程，但未能解决这一结构冗余问题。为此，我们提出BBRes框架，该框架将一种新颖的提前终止策略融入递归分支过程。通过使用专用多项式时间求解器识别并处理易于求解的子实例，BBRes有效避免了冗余分支步骤。此外，我们设计了一种与终止机制协同的自适应分支策略，从而实现了更优的理论最坏情况时间复杂度。为提升实际性能，我们提出了一种基于新型双图着色方法与最大流技术相结合的更紧上界，该方法与分支框架正交。实验表明，BBRes在大量数据集上取得相对于现有方法至少2倍的加速比。