Given a set $A$ of subsets of $\{1, \ldots n\}$ which is closed under taking subsets, let $X(A)$ be the corresponding simplicial complex, i.e. simplices of $X(A)$ are elements of the set $\bar A$, and gluing is induced by containment of subsets)

Consider the following computational problem

Input: a natural number $n$ and two sets $A$ and $B$ of subsets of $\{1,\ldots, n\}$, closed under taking subsets.

Problem: Are $X(A)$ and $X(B)$ isomorphic as simplicial complexes? (i.e. is there a bijection of $\{1,\ldots ,n}$ which bijectively sends faces of $X(A)$ to faces of $X(B)$?)

**Question:** I'm interested to know what algorithms are known for this problem. I'm specifically interested in worst running times in terms of $n$ alone. Please note that the size of the input can be exponential in $n$.

In principle $A$ and $B$ might consist of $2^n$ subsets, so this is a lower bound for the problem, because the algorithm needs to read the input.

On the other hand the trivial algorithm of checking each permutation takes at most $\mathcal O(2^n\cdot n!)$ steps.