Representing Matroids over the Reals is $\exists \mathbb R$-complete

TL;DR

This paper proves that determining whether a matroid can be represented over the real numbers is an $ ext{exists} ext{R}$-complete problem, highlighting its computational complexity even for rank 3 matroids.

Contribution

The paper establishes the $ ext{exists} ext{R}$-completeness of the matroid realizability problem over the reals, providing a formal complexity classification and a proof in the language of computer science.

Findings

Matroid realizability over reals is $ ext{exists} ext{R}$-complete.

The result holds even for matroids of rank 3.

Provides a formal proof in the language of computer science.

Abstract

A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $\varphi(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.