d-collapsibility is NP-complete for d>=4

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. 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.