Approximation Algorithm for Constrained $k$-Center Clustering: A Local Search Approach

2026-01-17

AI Summary

This study addresses the k-center clustering problem under instance-level cannot-link (CL) and must-link (ML) constraints, which significantly increases the approximation difficulty due to the added restrictions. To tackle this challenge, the authors propose a novel local search framework that reformulates the problem as a dominating matching set problem, achieving a 2-approximation ratio while respecting disjoint CL constraints. This work establishes, for the first time, that local search can attain a constant-factor approximation that is theoretically optimal under such constraints, thereby resolving an open question regarding the feasibility of constant-factor approximations via local search in this setting. Experimental results demonstrate that the proposed algorithm consistently outperforms existing baselines on both real-world and synthetic datasets, effectively balancing theoretical optimality with practical performance.

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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Clustering is a long-standing research problem and a fundamental tool in AI and data analysis. The traditional k-center problem, a fundamental theoretical challenge in clustering, has a best possible approximation ratio of 2, and any improvement to a ratio of 2 - {\epsilon} would imply P = NP. In this work, we study the constrained k-center clustering problem, where instance-level cannot-link (CL) and must-link (ML) constraints are incorporated as background knowledge. Although general CL constraints significantly increase the hardness of approximation, previous work has shown that disjoint CL sets permit constant-factor approximations. However, whether local search can achieve such a guarantee in this setting remains an open question. To this end, we propose a novel local search framework based on a transformation to a dominating matching set problem, achieving the best possible approximation ratio of 2. The experimental results on both real-world and synthetic datasets demonstrate that our algorithm outperforms baselines in solution quality.

Problem

• constrained k-center clustering
• local search
• cannot-link constraints
• must-link constraints
• approximation algorithm

Innovation

• constrained k-center clustering
• local search
• approximation algorithm
• cannot-link constraints
• dominating matching set