A Faster Branching Algorithm for the Maximum $k$-Defective Clique Problem
This work addresses a graph problem important for social and biological network analysis, but it appears incremental as it builds on existing methods with specific improvements.
The paper tackles the maximum k-defective clique problem by proposing a new branching algorithm with improved asymptotic running time and a new upper-bounding technique based on conflict relationships, and experiments show it outperforms state-of-the-art solvers on open benchmarks.
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.