Optimal curve straightening to a target vertex count is ∃R-complete, and isotopy realization spaces of curves are universal up to homotopy equivalence.
The Complexity of Tensor Rank
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abstract
We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by H{\aa}stad~\cite{H90}. The hardness proof also implies an algebraic universality result.
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Optimal Curve Straightening is $\exists\mathbb{R}$-Complete
Optimal curve straightening to a target vertex count is ∃R-complete, and isotopy realization spaces of curves are universal up to homotopy equivalence.