Obstructions to weak decomposability for simplicial polytopes

TL;DR

This paper demonstrates that certain simplicial polytopes are not weakly decomposable, indicating that decomposability cannot be used to establish polynomial bounds on polytope diameters, thus impacting approaches to the Hirsch conjecture.

Contribution

The paper refines previous analyses to show these polytopes are not weakly $O( oot d)$-decomposable, providing new obstructions to using decomposability for diameter bounds.

Findings

Certain simplicial polytopes are not weakly $O( oot d)$-decomposable

Decomposability cannot prove polynomial diameter bounds

Implications for the Hirsch conjecture

Abstract

Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these $d$-dimensional polytopes are not even weakly $O(\sqrt{d})$-decomposable. As a consequence, (weak) decomposability cannot be used to prove a polynomial version of the Hirsch conjecture.