On the Complexity of Multi-Parameterized Cluster Editing

TL;DR

This paper explores the computational complexity of a multi-parameterized version of the Cluster Editing problem, showing NP-hardness under certain conditions but also identifying cases where the problem can be solved efficiently.

Contribution

The study introduces a multi-parameterized formulation of Cluster Editing, proving NP-hardness in general but providing polynomial-time solutions when edge edits per vertex are limited relative to cluster size.

Findings

NP-hardness persists even with minimal edge edits per vertex

Polynomial-time solvability when edits per vertex are less than half the cluster size

Linear-size kernelization algorithm for small constant bounds

Abstract

The Cluster Editing problem seeks a transformation of a given undirected graph into a disjoint union of cliques via a minimum number of edge additions or deletions. A multi-parameterized version of the problem is studied, featuring a number of input parameters that bound the amount of both edge-additions and deletions per single vertex, as well as the size of a clique-cluster. We show that the problem remains NP-hard even when only one edge can be deleted and at most two edges can be added per vertex. However, the new formulation allows us to solve Cluster Editing (exactly) in polynomial time when the number of edge-edit operations per vertex is smaller than half the minimum cluster size. In other words, Correlation Clustering can be solved efficiently when the number of false positives/negatives per single data element is expected to be small compared to the minimum cluster size. As a byproduct, we obtain a kernelization algorithm that delivers linear-size kernels when the two edge-edit bounds are small constants.