Revisiting the Maximum Defective Clique Problem: Faster Branching and a Tighter Upper Bound

Abstract

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.