Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Abstract

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded K [ x, y ]-module M , where K is a field. The algorithm takes as input a short chain complex of free modules X f −→ Y g −→ Z such that M ∼ = ker g/ im f . It runs in time O ( | X | 3 + | Y | 3 + | Z | 3 ) and requires O ( | X | 2 + | Y | 2 + | Z | 2 ) memory, where | · | denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of M are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gr¨obner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.