# The Facets of the Bases Polytope of a Matroid and Two Consequences

## TL;DR

This paper characterizes the facets of a matroid's bases polytope using locked subsets, leading to polynomial algorithms for maximum-weight basis and uniformity testing in certain matroids.

## Contribution

It introduces the concept of locked subsets to describe polytope facets and develops polynomial-time algorithms for maximum-weight basis and uniformity testing.

## Key findings

- Facets of the bases polytope are described by locked subsets.
- Maximum-weight basis can be found in polynomial time for certain matroids.
- A matroid oracle for testing uniformity is proposed.

## Abstract

Let $M$ to be a matroid defined on a finite set $E$ and $L\subset E$. $L$ is locked in $M$ if $M|L$ and $M^*|(E\backslash L)$ are 2-connected, and $min\{r(L), r^*(E\backslash L)\} \geq 2$. In this paper, we prove that the nontrivial facets of the bases polytope of $M$ are described by the locked subsets. We deduce that finding the maximum--weight basis of $M$ is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the V\'amos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.