Shellability is hard even for balls

TL;DR

This paper proves that determining shellability and collapsibility is NP-hard for certain classes of triangulated complexes in dimensions three and higher, highlighting the computational difficulty of these topological properties.

Contribution

It establishes NP-hardness of shellability for triangulated d-balls with d≥3, and collapsibility for 3-complexes in 3-space, extending previous hardness results.

Findings

Shellability is NP-hard for triangulated d-balls when d≥3.

Collapsibility is NP-hard for 3-complexes embeddable in 3-space.

Shellability is NP-hard for 2-complexes embeddable in 3-space.

Abstract

The main goal of this paper is to show that shellability is NP-hard for triangulated d-balls (this also gives hardness for triangulated d-manifolds/d-pseudomanifolds with boundary) as soon as d is at least 3. This extends our earlier work with Goaoc, Pat\'akov\'a and Wagner on hardness of shellability of 2-complexes and answers some questions implicitly raised by Danaraj and Klee in 1978 and explicitly mentioned by Santamar\'ia-Galvis and Woodroofe. Together with the main goal, we also prove that collapsibility is NP-hard for 3-complexes embeddable in the 3-space, extending an earlier work of the second author and answering an open question mentioned by Cohen, Fasy, Miller, Nayyeri, Peng and Walkington; and that shellability is NP-hard for 2-complexes embeddable in the 3-space, answering another question of Santamar\'ia-Galvis and Woodroofe (in a slightly stronger form than what is given by the main result).