## Abstract

We present two algorithms in which constrained spectral clustering is implemented as unconstrained spectral clustering on a multi-layer graph where constraints are represented as graph layers. By using the Nystrom approximation in one of the algorithms, we obtain time and memory complexities which are linear in the number of data points regardless of the number of constraints. Our algorithms achieve superior or comparative accuracy on real world data sets, compared with the existing state-of-the-art solutions. However, the complexity of these algorithms is squared with the number of vertices, while our technique, based on the Nyström approximation method, has linear time complexity. The proposed algorithms efficiently use both soft and hard constraints since the time complexity of the algorithms does not depend on the size of the set of constraints.