  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~\cite{DeyCheng18}.
