Finding Short Paths on Simple Polytopes

We prove that computing a shortest monotone path to the optimum of a linear program over a simple polytope is NP-hard, thus resolving a 2022 open question of De Loera, Kafer, and Sanità. As a consequence, finding a shortest sequence of pivots to an optimal basis with the simplex method is NP-hard. In fact, we show this is NP-hard already for fractional knapsack polytopes. By applying an additional polyhedral construction, we show that computing the diameter of a simple polytope is NP-hard, resolving a 2003 open problem by Kaibel and Pfetsch. Finally, on the positive side we show that every polytope has a small, simple extended formulation for which a linear length path may be found between any pair of vertices in polynomial time building upon a result of Kaibel and Kukharenko.

Source: arXiv:2603.05482v1 - http://arxiv.org/abs/2603.05482v1 PDF: https://arxiv.org/pdf/2603.05482v1 Original Link: http://arxiv.org/abs/2603.05482v1