  A simplicial complex is d-collapsible if it can be reduced to an empty
complex by repeatedly removing (collapsing) a face of dimension at most d-1
that is contained in a unique maximal face. We prove that the algorithmic
question whether a given simplicial complex is d-collapsible is NP-complete for
d greater or equal to 4 and polynomial time solvable for d at most 2. As an
intermediate step, we prove that d-collapsibility can be recognized by the
greedy algorithm for d at most 2, but the greedy algorithm does not work for d
greater or equal 3. A simplicial complex is d-representable if it is the nerve
of a collection of convex sets in R^d. The main motivation for studying
d-collapsible complexes is that every d-representable complex is d-collapsible.
We also observe that known results imply that analogical algorithmic question
for d-representable complexes is NP-hard for d greater or equal to 2.
