Constrained clustering, such as $k$-means with instance-level Must-Link (ML) and Cannot-Link (CL) auxiliary information as the constraints, has been extensively studied recently, due to its broad applications in data science and AI. Despite some heuristic approaches, there has not been any algorithm providing a non-trivial approximation ratio to the constrained $k$-means problem. To address this issue, we propose an algorithm with a provable approximation ratio of $O\mathsf{}\mathrm{(}\log \mathsf{}k\mathrm{)}$ when only ML constraints are considered. We also empirically evaluate the performance of our algorithm on real-world datasets having artificial ML and disjoint CL constraints. The experimental results show that our algorithm outperforms the existing greedy-based heuristic methods in clustering accuracy.