37 research outputs found
Color Matching
We consider the problem of creating paint of a certain target color by mixing colorants. Although a large number of colorants are available, in practice it is only allowed to use a limited number. We focus on the problem of selecting the right subset of colorants
Relative rank axioms for infinite matroids
In a recent paper, Bruhn, Diestel, Kriesell and Wollan (arXiv:1003.3919)
present four systems of axioms for infinite matroids, in terms of independent
sets, bases, closure and circuits. No system of rank axioms is given. We give
an easy example showing that rank function of an infinite matroid may not
suffice to characterize it. We present a system of axioms in terms of relative
rank.Comment: The results in this paper have now been merged into arXiv:1003.391
Counting matroids in minor-closed classes
A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an -minor is
asymptotically small in case is one of the sparse paving matroids
, , , , or , thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without -minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure
Reconstructing a phylogenetic level-1 network from quartets
We describe a method that will reconstruct an unrooted binary phylogenetic
level-1 network on n taxa from the set of all quartets containing a certain
fixed taxon, in O(n^3) time. We also present a more general method which can
handle more diverse quartet data, but which takes O(n^6) time. Both methods
proceed by solving a certain system of linear equations over GF(2).
For a general dense quartet set (containing at least one quartet on every
four taxa) our O(n^6) algorithm constructs a phylogenetic level-1 network
consistent with the quartet set if such a network exists and returns an (O(n^2)
sized) certificate of inconsistency otherwise. This answers a question raised
by Gambette, Berry and Paul regarding the complexity of reconstructing a
level-1 network from a dense quartet set
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
An entropy argument for counting matroids
We show how a direct application of Shearers' Lemma gives an almost optimum
bound on the number of matroids on elements.Comment: Short note, 4 page
Representing some non-representable matroids
We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutte's representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field.
Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field.
Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the Matrix Tree theorem for this class
Confinement of matroid representations to subsets of partial fields
Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem.
A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields.
We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6.
Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al