Abstract
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 Vamos 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.
Highlights
Sets and their characterisitic vectors will not be distinguished
We prove that the nontrivial facets of the bases polytope of M are described by the locked subsets
We prove that maximum-weight basis problem (MWBP) is polynomial on E for polynomially locked classes of matroids, i.e., for any matroid M ∈ k
Summary
Sets and their characterisitic vectors will not be distinguished. We refer to Oxley [1] and Schrijver [2] about, respectively, matroids and polyhedra terminolgy and facts. ( X ) and * ( X ) are, respectively the class of bases of M | X and cobases of M * | X. The polyhedra Q ( M ) and P ( M ) are, respectively, the convex hulls of the independent sets and the bases of M. Suppose that M (and M * ) is 2-connected.
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