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：基于中心点的聚类在理论与实践中均吸引了大量研究兴趣。在许多实际应用中，输入数据常包含可用来改进聚类结果的背景知识。本研究基于广泛采用的k-中心聚类，将其输入背景知识建模为必须连接（ML）和不能连接（CL）约束集。然而，大多数聚类问题（包括k-中心）本质上是$\mathcal{NP}$-难的，而更复杂的带约束变体已知面临更严重的近似与计算障碍，严重限制了其适用性。通过采用反向支配集、线性规划（LP）整数多面体和LP对偶等一系列技术，我们首次为带约束k-中心问题提出了具有最优比率2的高效近似算法。我们还构建了竞争性基线算法，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。实验结果验证了我们的理论发现，并展示了算法在聚类代价、聚类质量和运行时间方面的显著优势。