Shellability is hard even for balls

TL;DR

The paper establishes that shellability is NP-hard for triangulated $d$-balls when $d\ge3$, and extends this hardness to $d$-manifolds with boundary as well as to embedded $2$-complexes in $3$-space. The proof hinges on a two-pronged reduction: first proving NP-hardness for collapsibility of 3-complexes embeddable in $\mathbb{R}^3$ and then thickening these gadgets into balls, preserving the equivalence between satisfiability of planar monotone rectilinear 3-SAT instances and shellability. Central to the construction are gadgets (bipyramid, 1-house, turbine, thick variants) and a carefully designed template that glues gadgets while controlling free faces and triangulations; a parallel shellability reduction is developed with analogous gadgets for 3-balls. The work also results in NP-hardness for shellability of embedded $2$-complexes, highlighting the depth of global obstructions to shellability and informing the landscape of 3-ball/3-sphere recognition. Overall, the paper advances our understanding of combinatorial topology's computational boundaries and provides a robust framework for hardness results via gadget-based reductions from SAT variants.

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, Patáková 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 Santamaría-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 Santamaría-Galvis and Woodroofe (in a slightly stronger form than what is given by the main result).

Figures (52)

• Figure 1: The bipyramid
• Figure 2: 1-house. (We are very thankful to our coauthors from gpptw19 for their approval to reuse and to further modify this drawing.)
• Figure 3: 1-house with our tree and crossing circles.
• Figure 4: 1-house with a our tree and crossing circles embedded in a cube.
• Figure 5: 1-house with our tree and crossing circles embedded in a cube after a homeomorphism.

Theorems & Definitions (52)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4: Shelling up
• Definition 5: Shelling = Shelling down
• Lemma 6
• proof
• Lemma 7: Lemma 9 in gpptw19
• Lemma 8
• proof