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

摘要

Let M be a matroid defined on a finite set E and L?⊂?E?. L is locked in M if??and ?are 2-connected, and . 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.