Efficiently Enumerating all Maximal Cliques with Bit-Parallelism

TL;DR

This paper introduces a bit-parallel algorithm with a greedy pivot strategy that significantly outperforms existing maximal clique enumeration algorithms on medium-density benchmark graphs, demonstrating notable speed improvements.

Contribution

The authors present a simple greedy pivot selection combined with bit-parallelism that outperforms state-of-the-art algorithms on standard benchmark instances for maximal clique enumeration.

Findings

Faster than the worst-case optimal algorithm on 60 out of 74 benchmarks

Solved 48 instances 1.2 to 2.2 times faster, and 12 instances 3.6 to 47.6 times faster

Achieved consistent speed improvements over the algorithm of Naudé

Abstract

The maximal clique enumeration (MCE) problem has numerous applications in biology, chemistry, sociology, and graph modeling. Though this problem is well studied, most current research focuses on finding solutions in large sparse graphs or very dense graphs, while sacrificing efficiency on the most difficult medium-density benchmark instances that are representative of data sets often encountered in practice. We show that techniques that have been successfully applied to the maximum clique problem give significant speed gains over the state-of-the-art MCE algorithms on these instances. Specifically, we show that a simple greedy pivot selection based on a fixed maximum-degree first ordering of vertices, when combined with bit-parallelism, performs consistently better than the theoretical worst-case optimal pivoting of the state-of-the-art algorithms of Tomita et al. [Theoretical Computer Science, 2006] and Naud\'e [Theoretical Computer Science, 2016]. Experiments show that our algorithm is faster than the worst-case optimal algorithm of Tomita et al. on 60 out of 74 standard structured and random benchmark instances: we solve 48 instances 1.2 to 2.2 times faster, and solve the remaining 12 instances 3.6 to 47.6 times faster. We also see consistent speed improvements over the algorithm of Naud\'e: solving 61 instances 1.2 to 2.4 times faster. To the best of our knowledge, we are the first to achieve such speed-ups compared to these state-of-the-art algorithms on these standard benchmarks.