Computing Bottleneck Distance for Multi-parameter Interval Decomposable Persistence Modules

Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For 1-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most n-parameter persistence modules, n>1, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called em n-parameter interval decomposable modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called em dimension distance that bounds it from below. An earlier version of this paper considered only the 2-parameter interval decomposable modules~citeDeyCheng18.