Graph-based data clustering: a quadratic-vertex problem kernel for s-Plex Cluster Vertex Deletion

TL;DR

This paper introduces the s-Plex Cluster Vertex Deletion problem, a graph clustering task that generalizes clique deletion by allowing clusters to be s-plexes, and develops efficient data reduction rules to create problem kernels.

Contribution

It presents the first fixed-parameter algorithms with provably effective data reduction rules for the s-Plex Cluster Vertex Deletion problem, extending clustering techniques.

Findings

Developed effective data reduction rules for s-Plex Cluster Vertex Deletion.

Established a problem kernel for the s-Plex Cluster Vertex Deletion.

Compared the parameter efficiency of s-Plex Cluster Vertex Deletion with s-Plex Editing.

Abstract

We introduce the s-Plex Cluster Vertex Deletion problem. Like the Cluster Vertex Deletion problem, it is NP-hard and motivated by graph-based data clustering. While the task in Cluster Vertex Deletion is to delete vertices from a graph so that its connected components become cliques, the task in s-Plex Cluster Vertex Deletion is to delete vertices from a graph so that its connected components become s-plexes. An s-plex is a graph in which every vertex is nonadjacent to at most s-1 other vertices; a clique is an 1-plex. In contrast to Cluster Vertex Deletion, s-Plex Cluster Vertex Deletion allows to balance the number of vertex deletions against the sizes and the density of the resulting clusters, which are s-plexes instead of cliques. The focus of this work is the development of provably efficient and effective data reduction rules for s-Plex Cluster Vertex Deletion. In terms of fixed-parameter algorithmics, these yield a so-called problem kernel. A similar problem, s-Plex Editing, where the task is the insertion or the deletion of edges so that the connected components of a graph become s-plexes, has also been studied in terms of fixed-parameter algorithmics. Using the number of allowed graph modifications as parameter, we expect typical parameter values for s-Plex Cluster Vertex Deletion to be significantly lower than for s-Plex Editing, because one vertex deletion can lead to a high number of edge deletions. This holds out the prospect for faster fixed-parameter algorithms for s-Plex Cluster Vertex Deletion.