Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology

TL;DR

This paper explores the structure of posets derived from simple polytopes' 1-skeleta, proposes a conjecture related to the nonrevisiting path property, and proves it in specific cases, with implications for linear programming efficiency.

Contribution

It introduces a conjecture linking poset topology and the nonrevisiting path property, proves it for certain polytopes, and connects these findings to the efficiency of the simplex method.

Findings

Proven join property for simple polytopes with lattice Hasse diagrams.

Conjecture that directed paths in these graphs do not revisit facets, implying simplex method efficiency.

Homotopy equivalence of order complexes to spheres or balls for certain lattices.

Abstract

Given any polytope $P$ and any generic linear functional ${\bf c} $, one obtains a directed graph $G(P,{\bf c})$ from the 1-skeleton of $P$ by orienting each edge $e(u,v)$ from $u$ to $v$ for ${\bf c} (u) < {\bf c} ( v)$. For $P$ a simple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice $L$, the join of any collection $S$ of elements which all cover a common element $u$ in $L$ is proven to equal the sink of the smallest face of $P$ containing $u$ and all of the elements of $S$. The author conjectures for such $G(P,{\bf c})$ that no directed path in $G(P,{\bf c})$ ever revisits any facet of $P$. This would imply for such $P$ and ${\bf c}$ that the simplex method for linear programming is efficient under all possible pivot rules. This conjecture is proven for 3-polytopes and for spindles. For simple polytopes in which $G(P,{\bf c})$ is the Hasse diagram of a lattice $L$, the order complex of each open interval in $L$ is proven homotopy equivalent to a ball or a sphere. Applications are given to the weak Bruhat order, the Tamari lattice, and the Cambrian lattices. This paper concludes with an appendix by Dominik Preu\ss proving the monotone Hirsch conjecture for $P$ a simple polytope and $G(P,{\bf c})$ the Hasse diagram of a lattice. This confirms one of the main consequences that the author's conjecture would have.