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.

翻译：基于中心的聚类在理论和实践中都引起了广泛的研究兴趣。在许多实际应用中，输入数据通常包含可用于改进聚类结果的背景知识。本文基于广泛采用的$k$-中心聚类，将其输入背景知识建模为必连(ML)和禁连(CL)约束集。然而，包括$k$-中心在内的大多数聚类问题本质上都是$\mathcal{NP}$-难的，而更为复杂的约束变体已知会遭受更严重的近似和计算障碍，这极大地限制了其适用性。通过采用包括反向支配集、线性规划(LP)整数多面体和LP对偶性在内的一系列技术，我们首次提出了具有最佳比率为2的约束$k$-中心的高效近似算法。我们还构建了具有竞争力的基线算法，并在多种真实数据集上对我们的近似算法进行了实证评估。结果验证了我们的理论发现，并证明了我们的算法在聚类成本、聚类质量和运行时间方面的巨大优势。