The Maximum k-Defective Clique Problem (MDCP) aims to find a maximum k-defective clique in a given graph, where a k-defective clique is a relaxation clique missing at most k edges. MDCP is NP-hard and finds many real-world applications in analyzing dense but not necessarily complete subgraphs. Exact algorithms for MDCP mainly follow the Branch-and-bound (BnB) framework, whose performance heavily depends on the quality of the upper bound on the cardinality of a maximum k-defective clique. The state-of-the-art BnB MDCP algorithms calculate the upper bound quickly but conservatively as they ignore many possible missing edges. In this paper, we propose a novel CoLoring-based Upper Bound (CLUB) that uses graph coloring techniques to detect independent sets so as to detect missing edges ignored by the previous methods. We then develop a new BnB algorithm for MDCP, called KD-Club, using CLUB in both the preprocessing stage for graph reduction and the BnB searching process for branch pruning. Extensive experiments show that KD-Club significantly outperforms state-of-the-art BnB MDCP algorithms on the number of solved instances within the cut-off time, having much smaller search tree and shorter solving time on various benchmarks.

翻译：最大k缺陷团问题旨在给定图中寻找最大k缺陷团，其中k缺陷团是至多缺失k条边的松弛团。该问题属于NP难问题，在分析稠密但非完全子图的众多实际应用中有重要价值。针对MDCP的精确算法主要遵循分支定界框架，其性能高度依赖于最大k缺陷团基数上界的质量。现有最先进的BnB MDCP算法虽能快速计算上界，但因忽略大量可能缺失的边而导致上界过于保守。本文提出一种新颖的基于图着色的上界，该上界利用图着色技术检测独立集，从而发现先前方法忽略的缺失边。进一步，我们基于CLUB开发了名为KD-Club的新型BnB算法，在预处理阶段的图缩减和BnB搜索过程中的分支剪枝两个环节均应用该上界。大量实验表明，在截止时间内，KD-Club在可解决实例数量上显著优于当前最先进的BnB MDCP算法，并且在各类基准测试中具有更小的搜索树规模和更短的求解时间。