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Finding Short Paths on Simple Polytopes

Alexander E. Black, Raphael Steiner

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 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.

Finding Short Paths on Simple Polytopes

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 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.
Paper Structure (5 sections, 22 theorems, 46 equations, 3 figures)

This paper contains 5 sections, 22 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Shortest Paths
  3. Diameters
  4. Rock Extensions
  5. Conclusions

Key Result

Theorem 1.1

$k$-Distance on simple polytopes is NP-hard.

Figures (3)

  • Figure 1: Depicted are the four different ways a hyperplane can slice two edges of a $2$-face of a hyper-cube, which gives rise to the notions (c), (d), (e), and (f) of adacency in Lemma \ref{['lem:edgetypes']}. Note there are truly six ways this can occur, but the remaining two correspond to swapping $i$ and $j$ for edges of type (d) and (e).
  • Figure 2: Depicted is the silo construction in three dimensions in the normal fan of the polytope. Namely, the outer triangle corresponds to the normal cone of the vertex being cut off. We visualize this as a triangle by slicing the cone with a plane. Then the siloing subdivides that slice. The basis exchange graph corresponds to the dual graph of the triangulation. In this picture it is already visible that two cells may be of distance $d=3$ away from each other as is the case for the highlighted cells on the right side of the picture.
  • Figure 3: Depicted is the graph $G_{d}$ for $d = 5$. Vertices of the same height (i.e., second coordinate) are pairwise adjacent. Otherwise, there is an edge from a vertex to the first vertex above it and below it that is in the graph.

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 30 more