Abstract
We prove the 1991 conjecture by Brightwell and Winkler that counting the number of linear extensions for posets of height two is #P-complete. We further extend this result to incidence posets of graphs.
Highlights
Counting linear extensions (#LE) of a finite poset is a fundamental problem in both Combinatorics and Computer Science
In 1991, Brightwell and Winkler showed that #LE is #P-complete [BW91]
They conjectured that the following problem is #P-complete:
Summary
Counting linear extensions (#LE) of a finite poset is a fundamental problem in both Combinatorics and Computer Science. Height two means that P has two levels, i.e. no chains of length 3 This problem has been open for 27 years, most recently reiterated in [Hub, LS17]. Our second result is an extension of Theorem 1. It was proposed recently by Lee and Skipper in [LS17], motivated by the optimization of nonlinear functions over the much-studied correlation polytope Output: The number e(IG) of linear extensions of the incidence poset IG. Brightwell and Winkler write: “We strongly suspect that Linear Extension Count for posets of height 2 is still #P-complete, but it seems that an entirely different construction is required to prove this” [BW91]. We refer to [MM11, Pap94] for notation, basic definitions and results in computational complexity
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