1.64-Approximation for Chromatic Correlation Clustering via Chromatic Cluster LP

TL;DR

This paper introduces a novel randomized 1.64-approximation algorithm for Chromatic Correlation Clustering, significantly improving previous approximation factors by extending LP relaxations to handle multiple relational colors.

Contribution

It develops a chromatic cluster LP relaxation and a combined rounding strategy, advancing approximation techniques for richer relational clustering problems.

Findings

Achieved a 1.64-approximation factor for CCC.

Extended cluster LP framework to chromatic settings.

Bypassed integrality gap limitations of standard LP.

Abstract

Chromatic Correlation Clustering (CCC) generalizes Correlation Clustering by assigning multiple categorical relationships (colors) to edges and imposing chromatic constraints on the clusters. Unlike traditional Correlation Clustering, which only deals with binary $(+/-)$ relationships, CCC captures richer relational structures. Despite its importance, improving the approximation for CCC has been difficult due to the limitations of standard LP relaxations. We present a randomized $1.64$-approximation algorithm to the CCC problem, significantly improving the previous factor of $2.15$. Our approach extends the cluster LP framework to the chromatic setting by introducing a chromatic cluster LP relaxation and an rounding algorithm that utilizes both a cluster-based and a greedy pivot-based strategy. The analysis bypasses the integrality gap of $2$ for the CCC version of standard LP and highlights the potential of the cluster LP framework to address other variants of clustering problems.