Shellability is NP-complete

TL;DR

This paper proves that determining shellability and k-decomposability of pure simplicial complexes are NP-hard problems for all dimensions greater than or equal to 2, resolving longstanding open questions.

Contribution

It establishes NP-completeness of shellability and k-decomposability decision problems for all relevant dimensions, answering questions from decades ago.

Findings

Shellability decision is NP-hard for all d ≥ 2.

k-decomposability decision is NP-hard for all d ≥ 2 and k ≥ 0.

Deciding CL-shellability of a poset is NP-hard.

Abstract

We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional, simplicial complex is $k$-decomposable is NP-hard. For $d \ge 3$, both problems remain NP-hard when restricted to contractible pure $d$-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.