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

TL;DR

This paper introduces a local search algorithm for constrained k-center clustering with cannot-link and must-link constraints, achieving the optimal approximation ratio of 2 and demonstrating superior performance in experiments.

Contribution

It presents a novel local search framework transforming the problem into a dominating matching set problem, achieving the best possible approximation ratio of 2.

Findings

The algorithm achieves an approximation ratio of 2.

Experimental results show the algorithm outperforms baselines.

The framework effectively handles background knowledge constraints.

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.