Halfspace Representations of Path Polytopes of Trees
Amer Goel, Aida Maraj, Alvaro Ribot
TL;DR
We address the problem of describing the path polytope $P_T$ of a tree $T$ via a minimal $\mathcal{H}$-representation. The authors develop an inductive framework using gluing of star trees and toric fiber products to express $P_T$ as a toric fiber product of subpolytopes, enabling an explicit facet description consisting of edgewise nonnegativity, a leaf-sum constraint $\sum_{e \in E_{\mathrm{leaf}}(T)} x_e = 2$, and internal-vertex inequalities; internal degree-3 nodes contribute fewer facets through the $\Delta_{3,2}$ geometry. They prove a dimension formula $\dim P_T = |E| - |\{u : \deg(u)=2\}| - 1$ and establish minimality of the representation by induction on the number of internal nodes, reducing cases with degree-2 internal nodes by tree collapse. This construction yields a scalable, verifiable $\mathcal{H}$-representation that can facilitate membership tests and has potential applications in phylogenetics, tropical geometry, and algebraic statistics where path parametrizations arise. The approach blends combinatorial tree decompositions with toric geometry to obtain explicit, minimal halfspace descriptions for path polytopes.
Abstract
Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics, tropical geometry, and algebraic statistics. We provide a minimal halfspace representation of these polytopes. The construction is made inductively using toric fiber products.