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

TL;DR

This work tackles the constrained $k$-center problem with instance-level cannot-link ($CL$) and must-link ($ML$) constraints, addressing the hardness induced by $CL$ constraints and the practical need to leverage background knowledge. It introduces a novel local-search framework that transforms CL/ML constraints into a dominating matching set (DMS) on an auxiliary $l$-partite graph, and refines centers via a single-swap free DMS (SF-DMS) to achieve the best possible $2$-approximation under disjoint CL constraints. The approach combines ML-aware aggregation with CL-driven reconfiguration, and the authors prove that the resulting center set has size $| ext{C}|\, ext{≤}\,k$, attaining a $2$-approximation; they also adapt the method to scenarios without exact knowledge of the optimal radius $opt$. Empirical evaluations on real-world and synthetic datasets demonstrate that LSCKC consistently outperforms baselines in clustering quality and adheres to the $2$-approximation bound, validating its practical applicability for constrained clustering tasks.

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 - ε 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.

Figures (2)

• Figure 1: Empirical approximation ratio on the synthetic dataset with $k = 50$. (The blue values indicate the average clustering performance for each algorithm with varying the number of constraints.)
• Figure 2: Cost of the constrained $k$-center clustering.

Theorems & Definitions (17)

• Lemma 1
• Lemma 2
• proof
• Definition 3
• Definition 4
• Definition 5
• Proposition 6
• Definition 7
• Lemma 8
• proof