Shellability is NP-complete
Résumé
We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is \textup{NP}-hard, hence \textup{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 \textup{NP}-hard. For $d \ge 3$, both problems remain \textup{NP}-hard when restricted to contractible pure $d$-dimensional complexes.