Chromatic correlation clustering via cluster LP

TL;DR

This work extends correlation clustering to a chromatic setting by formulating and solving a chromatic cluster LP, which is then rounded to an integral solution with cost guarantees. The authors introduce preclustering to reduce problem complexity and bound the admissible edges, enabling polynomial-time near-optimal solving via a bounded sub-cluster LP and Monte Carlo sampling. The main result is a $(2+\varepsilon)$-approximation for Chromatic Correlation Clustering, improving prior bounds and leveraging a combination of LP relaxation, randomized rounding, and hierarchical lifting methods. The approach also demonstrates the broader utility of cluster-LP techniques for chromatic variants and aligns with concurrent progress achieving tighter constants.

Abstract

Correlation Clustering is a fundamental clustering problem, and there has been a line of work on improving the approximation ratio for this problem in recent years. A key algorithmic component in these works is the cluster LP. Chromatic Correlation Clustering is an interesting generalization that has also been intensively studied. In light of success of the cluster LP in Correlation Clustering, it would be an interesting question whether the cluster LP can be used in Chromatic Correlation Clustering. We answer this question with affirmatives by presenting a $(2+\varepsilon)$-approximation algorithm for Chromatic Correlation Clustering using a chromatic cluster LP.

Theorems & Definitions (40)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Definition 4.1: Preclustered instance
• Definition 4.2: Respecting solution
• Theorem 4.3
• Lemma 4.4
• proof
• Lemma 4.5