The graph of implicit edge dependencies for indecomposability and beyond

Abstract

A polytope is called indecomposable if it cannot be expressed nontrivially as a Minkowski sum of other polytopes. Since Gale introduced the concept in 1954, several increasingly strong criteria have been developed to characterize indecomposability. In this paper, we introduce a new approach to indecomposability for frameworks and polytopes based on the graph of implicit edge dependencies, which records proportionalities between edge lengths across all deformations. This yields a new indecomposability criterion that unifies and generalizes most previous approaches, and has additional consequences in the study of deformation cones. As a main application, we construct new indecomposable deformed permutahedra that are not matroid polytopes. In 1970, Edmonds already noted the difficulty of characterizing the extreme rays of the submodular cone, equivalently, indecomposable deformed permutahedra. Matroid polytopes of connected matroids form a well-known family of such examples. We exhibit a new infinite family of indecomposable deformations of the permutahedron, not arising from matroid polytopes, obtained by suitable truncations of certain graphical zonotopes. We further demonstrate the scope of our methods through several additional applications. In particular, we refute a conjecture of Smilansky (1987) on the relation between the numbers of vertices and facets of indecomposable polytopes. Moreover, we obtain new bounds on the dimensions of deformation cones and we construct and analyze uniquely decomposable polytopes.