Clustering with Locally Bounded Ignorance

2026-05-13

AI Summary

This work addresses the computational hardness of correlation clustering on fuzzy-edge graphs—graphs where edges with weight zero induce a complex structure—showing that the problem admits no polynomial kernel when parameterized solely by the total cost budget $k$. To overcome this barrier, the authors introduce the degeneracy $d$ or the closure number $c$ of the fuzzy-edge graph as additional structural parameters. They establish, for the first time, that polynomial kernels exist under the combined parameters $k + d$ and $k + c$, thereby breaking through the limitations of single-parameter kernelization. Through a careful blend of parameterized complexity analysis, structural graph characterizations, and reduction techniques, the study not only yields positive kernelization results under mild structural restrictions but also identifies highly constrained scenarios where the problem remains intractable, thus sharpening the known complexity landscape of correlation clustering.

Abstract

In Correlation Clustering, the input is a graph $G=(V,E)$ with weight function $ω: {V \choose 2}\to Z$ 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.

Problem

• Correlation Clustering
• polynomial kernel
• fuzzy edge graph
• parameterized complexity
• degeneracy

Innovation

• Correlation Clustering
• polynomial kernel
• fuzzy edge graph
• degeneracy
• parameterized complexity