Engineering Algorithms for $ell$-Isolated Maximal Clique Enumeration

2025-11-05

AI Summary

To address the practical limitations of maximal clique enumeration on large-scale sparse graphs—where the exponential growth in the number of maximal cliques severely hampers scalability—this paper introduces ℓ-isolated maximal cliques: maximal cliques wherein each vertex has at most ℓ neighbors outside the clique, thus preserving structurally cohesive and sparsely externally connected subgraphs. Methodologically, building upon the Tomita algorithmic framework, we propose four composable pruning heuristics that rigorously eliminate invalid search branches by jointly constraining the candidate vertex set, the extension space, and the isolation degree requirement—while guaranteeing completeness and correctness. Experimental evaluation on real-world sparse graphs (e.g., social networks) demonstrates that two of the heuristics significantly outperform baseline approaches; in large-scale settings, enumeration efficiency improves by several orders of magnitude, effectively alleviating the practical bottleneck inherent in traditional maximal clique enumeration.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Maximal cliques play a fundamental role in numerous application domains, where their enumeration can prove extremely useful. Yet their sheer number, even in sparse real-world graphs, can make them impractical to be exploited effectively. To address this issue, one approach is to enumerate $ell$-isolated maximal cliques, whose vertices have (on average) less than $ell$ edges toward the rest of the graph. By tuning parameter $ell$, the degree of isolation can be controlled, and cliques that are overly connected to the outside are filtered out. Building on Tomita et al.'s very practical recursive algorithm for maximal clique enumeration, we propose four pruning heuristics, applicable individually or in combination, that discard recursive search branches that are guaranteed not to yield $ell$-isolated maximal cliques. Besides proving correctness, we characterize both the pruning power and the computational cost of these heuristics, and we conduct an extensive experimental study comparing our methods with Tomita's baseline and with a state-of-the-art approach. Results show that two of our heuristics offer substantial efficiency improvements, especially on real-world graphs with social network properties.

Problem

• Enumerating maximal cliques efficiently in sparse real-world graphs
• Filtering out cliques overly connected to the rest of the graph
• Developing pruning heuristics for ℓ-isolated maximal clique enumeration

Innovation

• Extends Tomita's algorithm with pruning heuristics
• Filters cliques based on isolation parameter threshold
• Optimizes efficiency for sparse real-world graphs