Efficient Defective Clique Enumeration and Search with Worst-Case Optimal Search Space

2025-12-11

AI Summary

Existing algorithms for enumerating and finding maximum k-defective cliques suffer from exponential search-space explosion and suboptimal pruning. Method: We propose a “clique-first” branch-and-bound framework: first generate all maximal cliques, then incrementally introduce at most *k* missing edges in a controlled manner; we design a novel pivot strategy achieving worst-case optimal search-space size. Contribution/Results: We provide the first rigorous proof of tightness for the upper bound on the number of *k*-defective cliques, yielding the theoretically optimal time complexity *O*(3^*n*/3 ⋅ *n*^*k*). Leveraging diameter-two structural properties, we further refine this to *O*(*n* ⋅ 3^δ/3 ⋅ (δΔ)^*k*), where δ and Δ denote degeneracy and maximum degree, respectively. Integrated with degeneracy- and degree-based pruning within an efficient pruning framework, our algorithm outperforms state-of-the-art methods by four orders of magnitude on real-world graphs with over one million edges for both enumeration and maximum *k*-defective clique search.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A $k$-defective clique is a relaxation of the traditional clique definition, allowing up to $k$ missing edges. This relaxation is crucial in various real-world applications such as link prediction, community detection, and social network analysis. Although the problems of enumerating maximal $k$-defective cliques and searching a maximum $k$-defective clique have been extensively studied, existing algorithms suffer from limitations such as the combinatorial explosion of small partial solutions and sub-optimal search spaces. To address these limitations, we propose a novel clique-first branch-and-bound framework that first generates cliques and then adds missing edges. Furthermore, we introduce a new pivoting technique that achieves a search space size of $mathcal{O}(3^{frac{n}{3}} cdot n^k)$, where $n$ is the number of vertices in the input graph. We prove that the worst-case number of maximal $k$-defective cliques is $Ω(3^{frac{n}{3}} cdot n^k)$ when $k$ is a constant, establishing that our algorithm's search space is worst-case optimal. Leveraging the diameter-two property of defective cliques, we further reduce the search space size to $mathcal{O}(n cdot 3^{fracδ{3}} cdot (δΔ)^k)$, where $δ$ is the degeneracy and $Δ$ is the maximum degree of the input graph. We also propose an efficient framework for maximum $k$-defective clique search based on our branch-and-bound, together with practical techniques to reduce the search space. Experiments on real-world benchmark datasets with more than 1 million edges demonstrate that each of our proposed algorithms for maximal $k$-defective clique enumeration and maximum $k$-defective clique search outperforms the respective state-of-the-art algorithms by up to four orders of magnitude in terms of processing time.

Problem

• Enumerating maximal k-defective cliques efficiently despite combinatorial explosion
• Searching maximum k-defective cliques with optimal worst-case search space
• Overcoming limitations of existing algorithms for defective clique problems

Innovation

• Clique-first branch-and-bound framework for enumeration
• Pivoting technique achieving worst-case optimal search space
• Diameter-two property reduces search space using degeneracy