Finding Short Paths on Simple Polytopes

TL;DR

This paper proves that finding shortest paths on simple polytopes and related problems like polytope diameter are NP-hard, but also shows that small extended formulations enable polynomial-time path finding.

Contribution

It resolves several open problems by proving NP-hardness of shortest path and diameter computations on simple polytopes, and introduces a positive result on extended formulations.

Findings

Computing shortest monotone paths on simple polytopes is NP-hard.

Determining the diameter of a simple polytope is NP-hard.

Small extended formulations allow polynomial-time shortest path computations.

Abstract

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 Sanit\`a. 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.