A central problem in topological data analysis is that of computing the homology of a given simplicial complex. Said complexes can have arbitrary large number of simplices, as can happen, for example, if the space is the Rips-Vietoris or Cech complex of a large data cloud. Thus, pre-processing the simplicial complex to get a smaller complex with the same homology groups and then applying the homology algorithm to the smaller one, has been an active research topic in the last years. In this survey, we discuss some recent papers that examine the complexity of this simplification via Discrete Morse Theory. This survey was prepared as a final project for a course on Computational Topology at The Ohio State University.