1.64-Approximation for Chromatic Correlation Clustering via Chromatic Cluster LP

2025-07-21

AI Summary

This paper studies the Color-Correlated Clustering (CCC) problem—a generalized correlation clustering model that jointly handles multi-class edge labels (colors) while enforcing color-consistency constraints. To overcome the large integrality gap and limited approximation ratio of standard linear programming (LP) relaxations for CCC, we propose the Chromatic Cluster LP relaxation framework. Our method integrates cluster-based randomized rounding with a greedy pivot strategy. This approach achieves the first significant improvement in the approximation ratio for CCC, reducing it from the previous best of 2.15 to 1.64—thereby breaking the theoretical bottleneck imposed by conventional LP relaxations. The result demonstrates the effectiveness and superiority of explicit cluster-structure modeling for complex constrained clustering problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Clustering
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Extends Correlation Clustering to multi-color edge relationships
• Overcomes limitations of standard LP relaxations for CCC
• Provides 1.64-approximation algorithm via chromatic cluster LP

Innovation

• Chromatic cluster LP relaxation for CCC
• Randomized 1.64-approximation algorithm
• Cluster-based and greedy pivot rounding