Generalized Graph Clustering: Recognizing (p,q)-Cluster Graphs

Abstract

Cluster Editing is a classical graph theoretic approach to tackle the problem of data set clustering: it consists of modifying a similarity graph into a disjoint union of cliques, i.e, clusters. As pointed out in a number of recent papers, the cluster editing model is too rigid to capture common features of real data sets. Several generalizations have thereby been proposed. In this paper, we introduce (p, q)-cluster graphs, where each cluster misses at most p edges to be a clique, and there are at most q edges between a cluster and other clusters. Our generalization is the first one that allows a large number of false positives and negatives in total, while bounding the number of these locally for each cluster by p and q. We show that recognizing (p, q)-cluster graphs is NP-complete when p and q are input. On the positive side, we show that (0, q)-cluster, (p, 1)-cluster, (p, 2)-cluster, and (1, 3)-cluster graphs can be recognized in polynomial time.

Graph-based data clustering with overlaps

“…Fourth, the polynomial-time approximability of our problems remains unexplored. Fifth and finally, it seems promising to study overlaps in the context of the more general correlation clustering problems (see [1]) or by relaxing the demand for (maximal) cliques in cluster graphs by the demand for some reasonably dense subgraphs (as recently considered in the context of clustering without overlaps [18,19,21]).…”

Graph-Based Data Clustering with Overlaps

“…Fourth, the polynomial-time approximability of our problems remains unexplored. Fifth and finally, it seems promising to study overlaps in the context of the more general correlation clustering problems (see [1]) or by relaxing the demand for (maximal) cliques in cluster graphs by the demand for some reasonably dense subgraphs (as recently considered in the context of clustering without overlaps [18,19,21]). …”

A Brief Survey of Fixed-Parameter Parallelism

“…Our approach thus proves that Cluster Editing is FPP. It would also be interesting to extend this algorithm further to other variants of Cluster Editing (see, for example, [31][32][33]).…”