Constructing discrete Morse functions

Morse theory has been considered a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Morse function has been defined on a manifold, then information about its topology can be deduced from its critical elements. The purpose of this work is to design an algorithm to define optimal discrete Morse functions on general cell complex, where optimality entails having the least number of critical elements. This problem is proven here to be MAX–SNP hard. However, we provide a linear algorithm that, for the case of 2–manifolds, always reaches optimality. Moreover, we proved various results on the structure of a discrete Morse function. In particular, we provide an equivalent representation by hyperforests. From this point of view, we designed a construction of discrete Morse functions for general cell complexes of arbitrary finite dimension. The resulting algorithm is quadratic in time and, although not guaranteed to be optimal, gives optimal answers in most of the practical cases.