In this paper we propose an algorithm that builds sparse decision DAGs (directed acyclic graphs) out of a list of features or base classifiers. The basic idea is to cast the DAG design task as a Markov decision process. Each instance can decide to use or to skip each base classifier, based on the current state of the classifier being built. The result is a sparse decision DAG where the base classifiers are selected in a data-dependent way. The development of algorithm was directly motivated by improving the traditional cascade design in applications where the computational requirements of classifying a test instance are as important as the performance of the classifier itself. Beside outperforming classical cascade designs on benchmark data sets, the algorithm also produces interesting deep structures where similar input data follows the same path in the DAG, and subpaths of increasing length represent features of increasing complexity.