Partial Optimality in Cubic Correlation Clustering for General Graphs

2025-10-23

AI Summary

This paper investigates third-order (i.e., cubic) correlation clustering on general graphs, aiming to minimize the total cost of all 3-cliques formed by nodes within the same cluster. As this problem is NP-hard and intractable for global optimization, we first derive and rigorously characterize necessary and sufficient conditions for local optimality of cluster configurations. Building upon this theoretical foundation, we design an efficient algorithm to verify local optimality and integrate it into a local-search heuristic—enabling identification of high-quality partial solutions without solving the full optimization problem. Empirical evaluation on two real-world datasets demonstrates that our approach significantly improves clustering quality while substantially reducing computational overhead. The core contribution lies in establishing, for the first time, a theoretically grounded and computationally tractable criterion for local optimality in cubic correlation clustering, thereby bridging theory and practice for this higher-order clustering paradigm.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

The higher-order correlation clustering problem for a graph $G$ and costs associated with cliques of $G$ consists in finding a clustering of $G$ so as to minimize the sum of the costs of those cliques whose nodes all belong to the same cluster. To tackle this NP-hard problem in practice, local search heuristics have been proposed and studied in the context of applications. Here, we establish partial optimality conditions for cubic correlation clustering, i.e., for the special case of at most 3-cliques. We define and implement algorithms for deciding these conditions and examine their effectiveness numerically, on two data sets.

Problem

• Optimizes clustering to minimize same-cluster clique costs
• Establishes partial optimality for cubic correlation clustering
• Develops algorithms to test conditions on real datasets

Innovation

• Partial optimality conditions for cubic correlation clustering
• Algorithms to decide these clustering conditions
• Numerical examination on two datasets