On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering

TL;DR

This paper investigates the complexity of clustering graphs into 2-clubs, proving NP-hardness and W[2]-hardness results, and develops practical algorithms that outperform standard solvers on biological data.

Contribution

It establishes the hardness of 2-Club Cluster Editing and analyzes kernelization limits for 2-Club Cluster Vertex Deletion, providing effective reduction rules and a competitive solver.

Findings

2-Club Cluster Editing is W[2]-hard.

2-Club Cluster Vertex Deletion is fixed-parameter tractable but has no polynomial kernel.

The developed solver outperforms CPLEX on biological data.

Abstract

Editing a graph into a disjoint union of clusters is a standard optimization task in graph-based data clustering. Here, complementing classic work where the clusters shall be cliques, we focus on clusters that shall be 2-clubs, that is, subgraphs of diameter two. This naturally leads to the two NP-hard problems 2-Club Cluster Editing (the allowed editing operations are edge insertion and edge deletion) and 2-Club Cluster Vertex Deletion (the allowed editing operations are vertex deletions). Answering an open question from the literature, we show that 2-Club Cluster Editing is W[2]-hard with respect to the number of edge modifications, thus contrasting the fixed-parameter tractability result for the classic Cluster Editing problem (considering cliques instead of 2-clubs). Then focusing on 2-Club Cluster Vertex Deletion, which is easily seen to be fixed-parameter tractable, we show that under standard complexity-theoretic assumptions it does not have a polynomial-size problem kernel when parameterized by the number of vertex deletions. Nevertheless, we develop several effective data reduction and pruning rules, resulting in a competitive solver, clearly outperforming a standard CPLEX solver in most instances of an established biological test data set.