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Are there any efficient algorithms to find the shortest homotopy between two paths in a $2$-complex?

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    $\begingroup$ I think that if we take a presentation complex of a finitely presented group for which the word problem is undecidable, then we won't have an algorithm. For sure if the input paths are not known to be homotopic or if the homotopy is not given, then we won't have an algorithm to decide that they are or construct a homotopy. I am not sure if something changes if a homotopy is also given as an input, but it looks like it won't work either. $\endgroup$
    – plop
    Aug 26, 2020 at 18:30

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