Exchange distance of basis pairs in split matroids

Abstract

The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a global understanding of the structure of matroid bases is a fundamental goal in matroid optimization.While studying the structure of symmetric exchanges, Gabow proposed the problem that any pair of bases admits a sequence of symmetric exchanges. A different extension of the exchange axiom was proposed by White, who investigated the equivalence of compatible basis sequences. Farber studied the structure of basis pairs, and conjectured that the basis pair graph of any matroid is connected. These conjectures suggest that the family of bases of a matroid possesses much stronger structural properties than we are aware of.In the present paper, we study the distance of basis pairs of a matroid in terms of symmetric exchanges. In particular, we give an upper bound on the minimum number of exchanges needed to transform a basis pair into another for split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. As a corollary, we verify the above mentioned long-standing conjectures for this large class. Being a subclass of split matroids, our result settles the conjectures for paving matroids as well.

Serial exchanges in matroids

“…The proof now follows by induction. In [1], the authors verify, among other things, the following conjecture of Gabow [5] for split matroids, a class which contains paving matroids. We observe that in the special case of Conjecture 5.1 where E(N ) is the union of two bases, then the conjecture implies that N has a cyclic ordering.…”

“…Perhaps the strongest result thus far can be found in [8] where it was shown that Conjecture 1.1 is true when r(M ) and |E(M )| are relatively prime. It follows from recent results in [1] on split matroids, a class which includes paving matroids, that the conjecture is true for paving matroids M where |E(M )| ≤ 2r(M ). Coupled with Theorem 1.2, we can replace 2r(M ) by 2r(M ) + 1 in this bound since |E(M )| and r(M ) are relatively prime when |E(M )| = 2r(M ) + 1.…”

Partitioning into common independent sets via relaxing strongly base orderability

“…An easy reasoning shows that the statement holds for strongly base orderable matroids. Apart from this, the conjecture was settled for graphic matroids [13,26,38], sparse paving matroids [10], matroids of rank at most 4 [30] and 5 [28], split matroids [7], and spikes [5]. In what follows, we show that (P+) implies a slightly weakened version of the conjecture.…”