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

TL;DR

This paper introduces BBRes, a novel framework for the maximum k-defective clique problem that combines early termination, specialized branching, and a tighter upper bound, leading to significant speedups.

Contribution

The paper proposes BBRes, a new recursive framework with early termination and a double graph coloring upper bound, improving both theoretical complexity and practical performance.

Findings

BBRes achieves at least 2X speedup over existing methods.

The double graph coloring upper bound improves pruning efficiency.

Extensive experiments validate the effectiveness of BBRes.

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.