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.
Karp,Reducibility among combinatorial problems, 50 Years of Integer Programming 1958-2008: from the Early Years to the State-of-the-Art, Springer, 2009, pp
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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.