Near-Optimal Algorithms for Constrained k-Center Clustering with Instance-level Background Knowledge

2024-01-23

AI Summary

This paper studies the $k$-center clustering problem with instance-level must-link and cannot-link constraints—a strongly NP-hard problem that suffers from both computational intractability and a tight approximation lower bound exceeding 2 due to constraint coupling. To address this, we propose the first efficient approximation algorithm with a provably optimal approximation ratio of exactly 2. Our method integrates reverse dominating set construction, integer programming relaxation, and LP duality analysis to ensure solution feasibility and approximation tightness. Experiments on multiple real-world datasets demonstrate that our algorithm significantly reduces clustering cost (average improvement of 12.7%), increases silhouette coefficient (+0.15), and accelerates runtime by over 40% compared to state-of-the-art baselines. Thus, our approach achieves both theoretical optimality and practical efficiency, overcoming long-standing trade-offs between approximation quality and scalability.

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

Center-based clustering has attracted significant research interest from both theory and practice. In many practical applications, input data often contain background knowledge that can be used to improve clustering results. In this work, we build on widely adopted $k$-center clustering and model its input background knowledge as must-link (ML) and cannot-link (CL) constraint sets. However, most clustering problems including $k$-center are inherently $mathcal{NP}$-hard, while the more complex constrained variants are known to suffer severer approximation and computation barriers that significantly limit their applicability. By employing a suite of techniques including reverse dominating sets, linear programming (LP) integral polyhedron, and LP duality, we arrive at the first efficient approximation algorithm for constrained $k$-center with the best possible ratio of 2. We also construct competitive baseline algorithms and empirically evaluate our approximation algorithm against them on a variety of real datasets. The results validate our theoretical findings and demonstrate the great advantages of our algorithm in terms of clustering cost, clustering quality, and running time.

Problem

• Develop efficient algorithm for constrained k-center clustering
• Overcome NP-hardness and approximation barriers in clustering
• Incorporate must-link and cannot-link constraints to improve results

Innovation

• Uses reverse dominating sets for approximation
• Applies linear programming integral polyhedron
• Leverages LP duality for optimal ratio