On the Complexity of 2-club Cluster Editing with Vertex Splitting

TL;DR

This paper investigates the computational complexity of editing graphs into s-clubs with vertex splitting, establishing NP-completeness and fixed-parameter tractability results, and providing polynomial algorithms for forests.

Contribution

It introduces the complexity analysis of s-Club Cluster Editing with vertex splitting, proving NP-hardness and fixed-parameter tractability, and offers polynomial solutions for forests.

Findings

Both problems are NP-Complete and APX-hard.

They are fixed-parameter tractable with respect to the number of edits.

s-Club Cluster Vertex Splitting is polynomial-time solvable on forests.

Abstract

Editing a graph to obtain a disjoint union of s-clubs is one of the models for correlation clustering, which seeks a partition of the vertex set of a graph so that elements of each resulting set are close enough according to some given criterion. For example, in the case of editing into s-clubs, the criterion is proximity since any pair of vertices (in an s-club) are within a distance of s from each other. In this work we consider the vertex splitting operation, which allows a vertex to belong to more than one cluster. This operation was studied as one of the parameters associated with the Cluster Editing problem. We study the complexity and parameterized complexity of the s-Club Cluster Edge Deletion with Vertex Splitting and s-Club Cluster Vertex Splitting problems. Both problems are shown to be NP-Complete and APX-hard. On the positive side, we show that both problems are Fixed-Parameter Tractable with respect to the number of allowed editing operations and that s-Club Cluster Vertex Splitting is solvable in polynomial-time on the class of forests.