Optimal curve straightening to a target vertex count is ∃R-complete, and isotopy realization spaces of curves are universal up to homotopy equivalence.
The nonnegative rank of a matrix: Hard problems, easy solutions
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abstract
Using elementary linear algebra, we develop a technique that leads to solutions of two widely known problems on nonnegative matrices. First, we give a short proof of the result by Vavasis stating that the nonnegative rank of a matrix is NP-hard to compute. This proof is essentially contained in the paper by Jiang and Ravikumar, who discussed this topic in different terms fifteen years before the work of Vavasis. Secondly, we present a solution of the problem of Cohen and Rothblum on rational nonnegative factorizations, which was posed in 1993 and remained open.
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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.