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Shellability is NP-complete

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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https://hal.inria.fr/hal-02050505
Contributor : Xavier Goaoc <>
Submitted on : Wednesday, February 27, 2019 - 11:10:11 AM
Last modification on : Friday, November 15, 2019 - 9:05:04 AM

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Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner. Shellability is NP-complete. Journal of the ACM (JACM), Association for Computing Machinery, In press, 66 (3), ⟨10.1145/3314024⟩. ⟨hal-02050505⟩

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