1,311 research outputs found
Exchange distance of basis pairs in split matroids
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.Comment: 17 page
On circuits and serial symmetric basis-exchange in matroids
The way circuits, relative to a basis, are affected as a result of exchanging
a basis element, is studied. As consequences, it is shown that three
consecutive symmetric exchanges exist for any two bases of a matroid, and that
a full serial symmetric exchange, of length at most 6, exists for any two bases
of a matroid of rank 5. A new characterization of binary matroids, related to
basis-exchange, is presented
On the number of matroids
We consider the problem of determining , the number of matroids on
elements. The best known lower bound on is due to Knuth (1974) who showed
that is at least . On the other hand, Piff
(1973) showed that , and it has
been conjectured since that the right answer is perhaps closer to Knuth's
bound.
We show that this is indeed the case, and prove an upper bound on that is within an additive term of Knuth's lower bound. Our proof
is based on using some structural properties of non-bases in a matroid together
with some properties of independent sets in the Johnson graph to give a
compressed representation of matroids.Comment: Final version, 17 page
- …