A More Relaxed Model for Graph-Based Data Clustering: s-Plex Cluster Editing

We introduce the s-Plex Cluster Editing problem as a generalization of the wellstudied Cluster Editing problem, both being NP-hard and both being motivated by graph-based data clustering. Instead of transforming a given graph by a minimum number of edge modifications into a disjoint union of cliques (this is Cluster Editing), the task in the case of s-Plex Cluster Editing is to transform a graph into a cluster graph consisting of a disjoint union of so-called splexes. Herein, an s-plex is a vertex set S inducing a subgraph in which every vertex has degree at least |S| − s. Cliques are 1-plexes. The advantage of s-plexes for s ≥ 2 is that they allow to model a more relaxed cluster notion (s-plexes instead of cliques), better reflecting inaccuracies of the input data. We develop a provably effective preprocessing based on data reduction (yielding a so-called problem kernel), a forbidden subgraph characterization of s-plex cluster graphs, and a depth-bounded search tree which is used to find optimal edge modifications sets. Altogether, this yields efficient algorithms in case of moderate numbers of edge modifications, this often being a reasonable assumption under a maximum parsimony model for data clustering.