A short proof of a min–max relation for the bases packing of a matroid

Abstract

Let [Formula: see text] be a finite set, and [Formula: see text] be a matroid defined on [Formula: see text]. Given [Formula: see text], we use the notations ([Formula: see text]-maximum bases packing for the first one): [Formula: see text] such that [Formula: see text] for any [Formula: see text], and [Formula: see text] for any basis [Formula: see text], and [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give a short proof for the known min–max relation [Formula: see text]. Moreover, we prove that the minimum [Formula: see text] can be restricted to single elements and semi locked subsets only. A subset [Formula: see text] is semi locked in [Formula: see text] if [Formula: see text] is closed and 2-connected, and [Formula: see text]. We deduce then a polynomial algorithm to compute [Formula: see text] in a large class of matroids by using a matroid oracle related to semi locked subsets.