Matroid base polytope decomposition

TL;DR

This paper studies how matroid base polytopes can be decomposed into simpler polytopes, focusing on hyperplane splits, and provides conditions for when such decompositions are possible or impossible.

Contribution

It characterizes when matroid base polytopes admit hyperplane splits, especially for direct sums, and proves non-existence results for binary matroids and hypercube 1-skeletons.

Findings

Hyperplane splits exist under certain conditions for matroid base polytopes.

No hyperplane split exists for binary matroids.

Matroids with hypercube 1-skeletons cannot be decomposed into smaller matroid base polytopes.

Abstract

Let $P(M)$ be the matroid base polytope of a matroid $M$. A {\em matroid base polytope decomposition} of $P(M)$ is a decomposition of the form $P(M) = \bigcup\limits_{i=1}^t P(M_{i})$ where each $P(M_i)$ is also a matroid base polytope for some matroid $M_i$, and for each $1\le i \neq j\le t$, the intersection $P(M_{i}) \cap P(M_{j})$ is a face of both $P(M_i)$ and $P(M_j)$. In this paper, we investigate {\em hyperplane splits}, that is, polytope decompositions when $t=2$. We give sufficient conditions for $M$ so $P(M)$ has a hyperplane split and characterize when $P(M_1 \oplus M_2)$ has a hyperplane split where $M_1 \oplus M_2$ denote the {\em direct sum} of matroids $M_1$ and $M_2$. We also prove that $P(M)$ has not a hyperplane split if $M$ is binary. Finally, we show that $P(M)$ has not a decomposition if its 1-skeleton is the {\em hypercube}.