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 ingeniously 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：摘要：最大k-缺陷团问题（MDCP）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。MDCP为NP难问题，在分析密集但不一定完备的子图中具有广泛实际应用。MDCP的精确算法主要遵循分支定界（BnB）框架，其性能严重依赖于最大k-缺陷团基数的上界质量。现有最先进的BnB MDCP算法虽然计算上界速度快，但由于忽略了许多可能的缺失边，导致上界估计过于保守。本文提出一种新颖的基于图着色的上界（CLUB），通过巧妙运用图着色技术检测独立集，从而识别先前方法忽略的缺失边。我们进而开发了一种新的MDCP BnB算法——KD-Club，在图约简的预处理阶段和分支剪枝的BnB搜索过程中均使用CLUB。大量实验表明，在截止时间内，KD-Club在求解实例数量上显著优于最先进的BnB MDCP算法，且在多个基准测试上具有更小的搜索树和更短的求解时间。