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arXiv · 1711.08436

Shellability is NP-complete

Xavier Goaoc·Pavel Paták·Zuzana Patáková·Martin Tancer·Uli Wagner
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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.

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Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner. 2018-01-24. Shellability is NP-complete. https://arxiv.org/abs/1711.08436

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