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

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 $M^{\ast }|(E\backslash L)$ are 2-connected, and $min\{r(L),r^{\ast }(E\backslash L)\}\ge 2$. 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 Vámos 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.

8 pages. arXiv admin note: text overlap with arXiv:1606.05384