Engineering Algorithms for $\ell$-Isolated Maximal Clique Enumeration

TL;DR

This paper introduces four pruning heuristics to efficiently enumerate $ ext{l}$-isolated maximal cliques, filtering out less relevant cliques and significantly improving computational performance on social network graphs.

Contribution

The paper presents novel pruning heuristics for $ ext{l}$-isolated maximal clique enumeration, extending Tomita's algorithm with proven correctness and efficiency improvements.

Findings

Two heuristics significantly reduce computation time.

Heuristics perform well on social network graphs.

Method outperforms baseline and state-of-the-art approaches.

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.