TITLE: Testing simplicial complexes for shellability
QUESTION [25 upvotes]: Question

Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable?

By efficient here I am willing to consider anything with smaller expected complexity than the exponential mess one gets by naively testing all possible orderings of maximal facets.
Background
Let $\Delta$ be a simplicial complex and for each simplex $\sigma \in \Delta$ let $\bar{\sigma}$ denote the subcomplex generated by $\sigma$ and all its faces. Fix an ordering of its maximal facets $F_1,\ldots,F_K$, pick some $k \in \lbrace 1,\ldots,K\rbrace$ and define $\Delta_k$ to be the subcomplex generated by $\bigcup_{1\leq j \leq k} F_j$, i.e., all facets up to and incluing the $k$-th one.
Definition: We call this ordering of maximal facets a shelling if the intersection $\overline{F_{k+1}} \cap \Delta_k$ is a simplicial complex of dimension $\dim (F_{k+1}) - 1$ for each $k \in \lbrace 1,\ldots,K-1\rbrace$.
In general, the complex $\Delta$ need not be a combinatorial manifold or have a uniform top dimension for its maximal facets. It is known that if $\Delta$ is shellable then there exists a shelling by maximal facets ordered so that the dimension is decreasing along the order. So one method to simplify the computational burden is to test only those orderings $F_1,\ldots,F_K$ of maximal facets so that $\dim F_i \geq \dim F_j$ whenever $i \leq j$, but of course in the worst case all these facets could have the same dimension.
Motivation
Shellability is an extremely useful notion in topological combinatorics: many interesting simplicial complexes and posets in this field turn out to be shellable. I refer you to the works of Anders Bjorner and others for details, see here or here or... Since every shellable complex is a wedge of spheres, establishing shellability leads to all sorts of interesting conclusions. Among other things, shellable complexes must lack torsion in homology of all dimensions.

REPLY [3 votes]: This unpublished article (collaboration welcome)
presents a systematic approach to decide shellability, which goes beyond  extending partial shellings (that can confirm but hardly disprove shellability). Furthermore, our method  (unlike Moriyama's algorithm) does not need the face-numbers. Instead of the $n!$ permutations of the $n$ facets we deal with certain admissible chains in certain posets of cardinality at most $n \cdot 2^n$ (which is $<< n!$). The shellability status of some matroid and some chessboard complexes with up to 24 facets is determined, or redetermined. Moreover the total number of shellings can be calculated. For instance the simplicial complex of all trees of the complete graph $K_4$ has exactly 722965625856 shellings. The previously known lower bound to the number of shellings was $6!=720$.