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 well-studied Cluster Editing problem; both are NP-hard and both are 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 $s$-plexes. 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 $s\geq2$ is that they allow us 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 modification sets. Altogether, this yields efficient algorithms in case of moderate numbers of edge modifications; this is often a reasonable assumption under a maximum parsimony model for data clustering.