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



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



algorithms for comparing two simplicial complexes
An exp(O(n)) algorithm is given in "Hypergraph isomorphism and structural equivalence of boolean functions", Eugene M. Luks, STOC 1999.


