Acyclic Partial Matchings for Multidimensional Persistence: Algorithm and Combinatorial Interpretation

Given a simplicial complex and a vector-valued function on its vertices, we present an algorithmic construction of an acyclic partial matching on the cells of the complex compatible with the given function. This implies the construction can be used to build a reduced filtered complex with the same multidimensional persistent homology as of the original one filtered by the sublevel sets of the function. The correctness of the algorithm is proved, and its complexity is analyzed. A combinatorial interpretation of our algorithm based on the concept of a multidimensional discrete Morse function is introduced for the first time in this paper. Numerical experiments show a substantial rate of reduction in the number of cells achieved by the algorithm.