4 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
Rainbow bases in matroids
Recently, it was proved by B\'erczi and Schwarcz that the problem of
factorizing a matroid into rainbow bases with respect to a given partition of
its ground set is algorithmically intractable. On the other hand, many special
cases were left open.
We first show that the problem remains hard if the matroid is graphic,
answering a question of B\'erczi and Schwarcz. As another special case, we
consider the problem of deciding whether a given digraph can be factorized into
subgraphs which are spanning trees in the underlying sense and respect upper
bounds on the indegree of every vertex. We prove that this problem is also
hard. This answers a question of Frank.
In the second part of the article, we deal with the relaxed problem of
covering the ground set of a matroid by rainbow bases. Among other results, we
show that there is a linear function such that every matroid that can be
factorized into bases for some can be covered by rainbow
bases if every partition class contains at most 2 elements
Reconfiguration of basis pairs in regular matroids
In recent years, combinatorial reconfiguration problems have attracted great
attention due to their connection to various topics such as optimization,
counting, enumeration, or sampling. One of the most intriguing open questions
concerns the exchange distance of two matroid basis sequences, a problem that
appears in several areas of computer science and mathematics. In 1980, White
proposed a conjecture for the characterization of two basis sequences being
reachable from each other by symmetric exchanges, which received a significant
interest also in algebra due to its connection to toric ideals and Gr\"obner
bases. In this work, we verify White's conjecture for basis sequences of length
two in regular matroids, a problem that was formulated as a separate question
by Farber, Richter, and Shan and Andres, Hochst\"attler, and Merkel. Most of
previous work on White's conjecture has not considered the question from an
algorithmic perspective. We study the problem from an optimization point of
view: our proof implies a polynomial algorithm for determining a sequence of
symmetric exchanges that transforms a basis pair into another, thus providing
the first polynomial upper bound on the exchange distance of basis pairs in
regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on
the serial symmetric exchange property of matroids for the regular case.Comment: 28 pages, 6 figure