Shellability is NP-complete

Shellings from relative shellings, with an application to NP-completeness

“…This construction is originally due to Malgouyres and Francés in [21]. As observed in [12,Remark 9], there is also a somewhat more concrete construction due to Hachimori [13] (and extended to arbitrary dimensions by Adiprasito, Benedetti and Lutz [2,Theorem 2.3]).…”

“…See also [17]. This problem sat open for 40 years, until Goaoc, Paták, Patáková, Tancer and Wagner, in a significant recent advance, verified the problem to be NP-complete: Theorem 1.1 (Goaoc, Paták, Patáková, Tancer, and Wagner [12,Theorem 1]). SHELLABILITY is NP-complete, even when restricted to 2-dimensional simplicial complexes.…”

“…The proof is by polynomial reduction from 3SAT. As is typical in such a reduction, the construction in [12] proceeds by building "choice gadgets" (corresponding to variables in a 3SAT instance), "constraint gadgets" (corresponding to clauses), and some other needed gadgets for consistency. As an essential building block, these authors require simplicial complexes that are shellable, but where the shelling order is 'rigid' in a certain precise sense.…”

“…One such building block is a shellable complex with a single free face τ ; in such a complex, every shelling order must end with the facet containing τ . The authors of [12] use a construction based on Bing's house with 2 rooms. This construction is originally due to Malgouyres and Francés in [21].…”

“…Another building block in [12] is a shellable complex with 3 nonadjacent free faces, and satisfying some other conditions needed for gluing it to a larger construction. The authors in [12] roughly sketch a construction based on Bing's house for the latter building block, but do not provide all details. They instead give a reference to another paper of Tancer [30,Section 4], which provides more details of the construction.…”