Enumerative problems for arborescences and monotone paths on polytope graphs

TL;DR

This paper investigates the combinatorial complexity of arborescences and monotone paths on polytope graphs, providing bounds based on polytope dimensions and vertices or facets, with implications for geometric combinatorics and optimization.

Contribution

It introduces bounds on the number of $f$-arborescences, $f$-monotone paths, and the diameter of their graph structures for polytopes, advancing understanding of their combinatorial properties.

Findings

Bounds on the number of $f$-arborescences

Bounds on the number of $f$-monotone paths

Bounds on the diameter of the $f$-monotone path graph

Abstract

Every generic linear functional $f$ on a convex polytope $P$ induces an orientation on the graph of $P$. From the resulting directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$, as well as a natural graph structure on the vertex set of $f$-monotone paths. These concepts are important in geometric combinatorics and optimization. This paper bounds the number of $f$-arborescences, the number of $f$-monotone paths, and the diameter of the graph of $f$-monotone paths for polytopes $P$ in terms of their dimension and number of vertices or facets.