Indecomposability of 0/1-polytopes

TL;DR

This paper proves that every 0/1-polytope can be uniquely decomposed into indecomposable components, which are orthogonal and form a Cartesian product, with applications to various combinatorial polytopes.

Contribution

It establishes the unique Minkowski decomposition of 0/1-polytopes into indecomposable parts and provides criteria for indecomposability in several classes of polytopes.

Findings

Every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes.

Indecomposable summands lie in pairwise orthogonal subspaces.

Every nontrivial factorization of a multi-affine polynomial is in disjoint variable sets.

Abstract

We prove that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspaces. Thus, every 0/1-polytope is the Cartesian product of indecomposable 0/1-polytopes. As applications, we obtain uniform combinatorial indecomposability criteria for order and chain polytopes, matroid polytopes, stable set and clique polytopes, edge polytopes, flow polytopes, and 2-level/compressed polytopes. We also show that every nontrivial factorization of a multi-affine polynomial is a product of multi-affine polynomials in disjoint sets of variables.