Clustering with Locally Bounded Ignorance

TL;DR

This paper investigates the complexity of Correlation Clustering, showing it admits polynomial kernels under certain structural parameters of the fuzzy edge graph, and also establishes hardness results for restricted cases.

Contribution

It introduces new polynomial kernel results based on the structure of the fuzzy edge graph and demonstrates hardness in restricted structural settings.

Findings

Polynomial kernel when parameterized by k+d, where d is the degeneracy of the fuzzy edge graph.

Polynomial kernel when parameterized by k+c, where c is the closure of the fuzzy edge graph.

Hardness results for cases with restricted structure of the graph induced by edges and nonedges.

Abstract

In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $\omega: {V \choose 2}\to Z$ and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most $k$ is FPT with respect to $k$ but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by $k+d$, where $d$ is the degeneracy of the fuzzy edge graph, and when parameterized by $k+c$, where $c$ is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.