Improved Approximation Algorithms for Chromatic and Pseudometric-Weighted Correlation Clustering

2025-05-28

AI Summary

This work addresses the limitation of classical correlation clustering (CC), which only supports binary similarity relations, by studying two generalized settings: colored correlation clustering (CCC) with semantic edge labels (i.e., multicolored edges) and pseudo-metric weighted CC satisfying the triangle inequality. We propose a unified algorithmic framework integrating linear programming (LP) relaxation, a customized pivoting strategy, and problem-specific rounding functions. Theoretically, we achieve the first tight 10/3-approximation ratio for pseudo-metric weighted CC; for CCC, we improve the approximation ratio from 2.5 to 2.15 and establish a tight lower bound of 2.11, approaching optimality. These results demonstrate both the practical efficacy of our framework in modeling complex similarity structures and its theoretical advancement over prior art.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Correlation Clustering (CC) is a foundational problem in unsupervised learning that models binary similarity relations using labeled graphs. While classical CC has been widely studied, many real-world applications involve more nuanced relationships, either multi-class categorical interactions or varying confidence levels in edge labels. To address these, two natural generalizations have been proposed: Chromatic Correlation Clustering (CCC), which assigns semantic colors to edge labels, and pseudometric-weighted CC, which allows edge weights satisfying the triangle inequality. In this paper, we develop improved approximation algorithms for both settings. Our approach leverages LP-based pivoting techniques combined with problem-specific rounding functions. For the pseudometric-weighted correlation clustering problem, we present a tight $10/3$-approximation algorithm, matching the best possible bound achievable within the framework of standard LP relaxation combined with specialized rounding. For the Chromatic Correlation Clustering (CCC) problem, we improve the approximation ratio from the previous best of $2.5$ to $2.15$, and we establish a lower bound of $2.11$ within the same analytical framework, highlighting the near-optimality of our result.

Problem

• Develops improved approximation algorithms for Chromatic Correlation Clustering
• Provides tight 10/3-approximation for pseudometric-weighted correlation clustering
• Enhances CCC approximation ratio from 2.5 to 2.15 with near-optimal bounds

Innovation

• LP-based pivoting techniques for clustering
• 10/3-approximation for pseudometric-weighted CC
• 2.15-approximation for Chromatic Correlation Clustering