Computing shortest monotone paths on simple polytopes is NP-hard, implying NP-hardness for shortest simplex pivot sequences and polytope diameters, with a polynomial-time result via small simple extended formulations.
Santos,A counterexample to the Hirsch conjecture, Annals of Mathematics (2012), 383–412
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Finding Short Paths on Simple Polytopes
Computing shortest monotone paths on simple polytopes is NP-hard, implying NP-hardness for shortest simplex pivot sequences and polytope diameters, with a polynomial-time result via small simple extended formulations.