Kernekoncepter
The core message of this article is that the problems of Sigma Clique Cover (SCC), Cluster Vertex Splitting (CVS), and Cluster Editing with Vertex Splitting (CEVS) are all NP-complete, but CVS admits a linear-size problem kernel.
Resumé
The article studies the computational complexity and algorithms for overlapping graph clustering problems, where the clusters can overlap. It focuses on three main problems:
Sigma Clique Cover (SCC): Given a graph G, find a covering of the edges of G by induced cliques, while minimizing the total number of times vertices are covered.
The authors show that SCC is NP-complete.
Cluster Vertex Splitting (CVS): Given a graph G and an integer k, determine if G can be transformed into a cluster graph (disjoint union of cliques) by performing at most k vertex-splitting operations.
The authors establish an equivalence between SCC and CVS, showing that CVS is also NP-complete.
However, they prove that CVS admits a linear-size problem kernel, meaning it can be efficiently preprocessed to an equivalent instance with O(k) vertices.
Cluster Editing with Vertex Splitting (CEVS): Given a graph G and an integer k, determine if G can be transformed into a cluster graph by performing at most k edge additions, edge deletions, and vertex-splitting operations.
The authors show that CEVS is NP-hard, building on their NP-hardness result for SCC and using a critical-clique lemma.
The article also discusses the relationship between these overlapping clustering problems and the well-studied Cluster Editing (CE) and Edge Clique Cover (ECC) problems, which do not allow for overlapping clusters.