Computational Complexity of the Interleaving Distance

TL;DR

This paper investigates the computational complexity of the interleaving distance in topological data analysis, establishing NP-hardness results and connections to matrix invertibility and graph isomorphism problems.

Contribution

It provides the first complexity bounds for computing the interleaving distance across various categories of persistence modules.

Findings

NP-hardness of computing interleaving distance for vector space modules

Interleaving distance in multidimensional persistence relates to matrix invertibility

Isomorphism problem for Reeb graphs is graph isomorphism complete

Abstract

The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.