Let $Y$ be a $1$-reduced finite simplicial set, $\Sigma^n$ be a simplicial model of the $n$-sphere and $f: \Sigma^n\to Y$. We further know that $f$ is nullhomotopic (this can be algorithmically verified via obstruction theory and Postnikov towers such as in this paper).
Our goal is to construct an explicit simplicial extension $F: B^{n+1}\to Y$ of $f$ such that $B^{n+1}$ is a triangulation of the $(n+1)$-disc and $\partial B^{n+1}$ is a simplicial subdivision of $\Sigma^n$.
Such algorithm clearly exists, as one could add one vertex to $\Sigma^n$ to create a simplicial model of $B^{n+1}$, then do iterative barycentric subdivisions and "try" all simplicial maps to $Y$ and eventually succeed.
Is there a more feasible algorithm with, let's say, polynomial complexity?
If we replace $Y$ via its Postnikov tower $P_{n+1}$ and replace $f$ by $(\varphi f)$ where $\varphi: Y\to P_{n+1}$ is $n$-connected, then we can construct an extension $B^{n+1}\to P_{n+1}$ explicitely, see, for example, this paper. However, we don't see how to find a nullhomotopy/extension to $Y$, which in general is not Kan.
