Switching regression problems are particular in that the data is assumed to be generated by a collection of processes and a switching mechanism activating one and only one of these processes at a time. This occurs for instance when modeling a system with different operating modes.
In this case, the problem is twofod: we must classify the data points to different modes and estimate a model for each one of these modes. All the difficulty comes from the intrinsic inseparability of these two subproblems: the classification influences the model estimation (since the local data set assigned to a mode changes with the classification) while the classification itself also depends on the model.
A simple approach to this problem is taken by the $K$-LinReg algorithm.