A $k$-defective clique of an undirected graph $G$ is a subset of its vertices that induces a nearly complete graph with a maximum of $k$ missing edges. The maximum $k$-defective clique problem, which asks for the largest $k$-defective clique from the given graph, is important in many applications, such as social and biological network analysis. In the paper, we propose a new branching algorithm that takes advantage of the structural properties of the $k$-defective clique and uses the efficient maximum clique algorithm as a subroutine. As a result, the algorithm has a better asymptotic running time than the existing ones. We also investigate upper-bounding techniques and propose a new upper bound utilizing the \textit{conflict relationship} between vertex pairs. Because conflict relationship is common in many graph problems, we believe that this technique can be potentially generalized. Finally, experiments show that our algorithm outperforms state-of-the-art solvers on a wide range of open benchmarks.

翻译：在无向图$G$中，$k$-缺陷团是指其顶点的一个子集，该子集诱导出一个最多缺失$k$条边的近似完全图。最大$k$-缺陷团问题要求从给定图中找出规模最大的$k$-缺陷团，该问题在社会网络与生物网络分析等众多应用中具有重要意义。本文提出一种新的分支算法，该算法充分利用$k$-缺陷团的结构特性，并以高效的最大团算法作为子程序。因此，该算法具有优于现有算法的渐近运行时间。我们还研究了上界技术，并提出一种利用顶点对间冲突关系的新型上界方法。由于冲突关系在图论问题中普遍存在，我们相信该技术具有潜在的普适性。最终，实验结果表明我们的算法在广泛的公开基准测试中优于当前最先进的求解器。