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≥ 4 and polynomial time solvable for d≤2.

As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for d< =2, but the greedy algorithm does not work for d> =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 d-representability is NP-hard to decide for d ≥ 2.