Efficient Algorithms and Complexity for Overlapping Graph Clustering via Vertex Splitting

Core Concepts

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.

Abstract

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.

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The Complexity of Cluster Vertex Splitting and Company

by Alex... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2309.00504.pdf

Deeper Inquiries

What are some real-world applications where overlapping graph clustering with small overlaps is particularly relevant

Overlapping graph clustering with small overlaps is particularly relevant in various real-world applications, including social network analysis, biological network analysis, and image segmentation. In social networks, individuals can belong to multiple communities or groups simultaneously, leading to overlapping clusters. In biological networks, proteins can participate in multiple pathways or functions, resulting in overlapping clusters. Image segmentation tasks may involve objects or regions that share common features, leading to overlapping clusters in the segmentation results. By allowing for overlaps in clustering, these applications can better capture the complex and interconnected nature of the data.

How do the complexity and algorithmic results for overlapping clustering problems compare to the non-overlapping variants, such as Cluster Editing and Edge Clique Cover

The complexity and algorithmic results for overlapping clustering problems differ from non-overlapping variants such as Cluster Editing and Edge Clique Cover. Overlapping clustering problems introduce additional challenges due to the need to handle shared elements between clusters. The NP-completeness of problems like Cluster Vertex Splitting in the context of overlapping clusters indicates that finding optimal solutions is computationally hard. In contrast, non-overlapping clustering problems focus on disjoint clusters, which may have different algorithmic properties and complexity results. For example, the Edge Clique Cover problem deals with covering all edges with induced cliques, which is distinct from the challenges posed by overlapping clusters.

Can the linear-size problem kernel for Cluster Vertex Splitting be further improved, or are there inherent limitations to the size of the kernel

While the linear-size problem kernel for Cluster Vertex Splitting is a significant achievement, further improvements may be possible depending on the specific characteristics of the problem. The size of the problem kernel is influenced by the structural properties of the input instances and the reduction rules applied. In some cases, there may be inherent limitations to reducing the problem size further without compromising the optimality of the solution. However, by exploring additional reduction techniques, refining the analysis of critical cliques, or considering alternative approaches, it may be feasible to enhance the problem kernel for Cluster Vertex Splitting. Further research and experimentation could provide insights into optimizing the problem kernel size.