Indecomposability and beyond via the graph of edge dependencies

TL;DR

This paper introduces a new criterion for indecomposability of polytopes using the graph of edge dependencies, unifying previous methods and expanding applications to deformation cones and new indecomposable polytopes.

Contribution

It presents a novel indecomposability criterion based on edge dependency graphs, generalizing prior approaches and enabling the construction of new indecomposable polytopes beyond matroid polytopes.

Findings

Developed a unified criterion for indecomposability using edge dependency graphs.

Constructed new indecomposable deformed permutahedra not arising from matroids.

Refuted a conjecture on the vertex-facet ratio of indecomposable polytopes.

Abstract

A polytope is called indecomposable if it cannot be expressed (non-trivially) as a Minkowski sum of other polytopes. Since the concept was introduced by Gale in 1954, several increasingly strong criteria have been developed to characterize indecomposability. In this paper, we introduce a new indecomposability criterion that unifies and generalizes most existing approaches. The key new ingredient is the graph of (implicit) edge dependencies, which records proportionalities between edge lengths across deformations and has broader applications in the study of deformation cones of polytopes, beyond indecomposability. As a main application, we construct new indecomposable deformed permutahedra that are not matroid polytopes. In 1970, Edmonds posed the problem of characterizing the extreme rays of the submodular cone, equivalently, indecomposable deformed permutahedra. Matroid polytopes from connected matroids form a well-known family of such examples. We exhibit a new infinite family of indecomposable deformations of the permutahedron, disjoint from matroid polytopes, obtained by suitable truncations of vertices of certain graphical zonotopes. We further demonstrate the scope of our methods through several additional applications. In particular, we refute a conjecture by Smilansky (1987) asserting that indecomposable polytopes must have relatively few vertices compared to their number of facets. We also obtain new bounds on the dimensions of deformation cones and characterize certain of their extreme rays, introduce parallelogramic Minkowski sums whose deformation cones factor as products, and provide new constructions of indecomposable polytopes via truncations and stackings.