Exact Algorithms for Finding Well-Connected 2-Clubs in Real-World Graphs: Theory and Experiments

TL;DR

This paper introduces exact algorithms for identifying well-connected 2-clubs in large real-world graphs, addressing their computational complexity and demonstrating superior performance over existing methods through extensive experiments.

Contribution

The paper develops the first exact combinatorial algorithms for well-connected 2-clubs, incorporating efficient data reduction and outperforming previous algorithms on real-world data.

Findings

Algorithms significantly outperform existing methods on sparse real-world graphs.

Finding well-connected 2-clubs is NP-hard, but effective exact algorithms are feasible.

Empirical results demonstrate practical efficiency and scalability.

Abstract

Finding large "cliquish" subgraphs is a central topic in graph mining and community detection. A popular clique relaxation are 2-clubs: instead of asking for subgraphs of diameter one (these are cliques), one asks for subgraphs of diameter at most two (these are 2-clubs). A drawback of the 2-club model is that it produces star-like hub-and-spoke structures as maximum-cardinality solutions. Hence, we study 2-clubs with the additional constraint to be well-connected. More specifically, we investigate the algorithmic complexity for three variants of well-connected 2-clubs, all established in the literature: robust, hereditary, and "connected" 2-clubs. Finding these more cohesive 2-clubs is NP-hard; nevertheless, we develop an exact combinatorial algorithm, extensively using efficient data reduction rules. Besides several theoretical insights we provide a number of empirical results based on an engineered implementation of our exact algorithm. In particular, the algorithm significantly outperforms existing algorithms on almost all (sparse) real-world graphs we considered.