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About 3,429 papers for “Matroid”

Matroid Covering

N. Young · eCommons (Cornell University) · 1994

Matroid Covering

Cited by 0Open access

Matroid Theory

AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory <1995, University of Washington>, Joseph E. Bonin, Oxley, J. G. et al. · Contemporary mathematics - American Mathematical Society · 1996

Cited by 4

Matroid designs

P. Young, J. Edmonds · Journal of Research of the National Bureau of Standards Section B Mathematical Sciences · 1973

Matroids are investigated in which equicardinality conditions are imposed on the flats. S uch matroids are shown to be closely related to ce rt ain types of BIB D's . Ex iste nce and unique ness theorems for thes e structures are derive d, toge th er with a bound ed ness crit eri on on th eir rank. Several classes are exhibited , in cludin g finite proj ective and affine geo metri es, ce rt ain tdesigns (S tein er sys te ms) and so-called trivi oid s. By vi e win g ce rtain t -des igns as matroids, new ways of constructin g BIBD's are deriv ed. Three ne w seri es of 3designs a nd two new 4designs are obtained by th ese methods. A matroid analysis of th e 5(24,8,1) desi gn of Witt is prese nte d, and exam pl es are obtained from it of matroids havin g e qui cardin al hyperpl anes but not e qui cardin a l fl ats in lower rank s. Several ge neral co nj ectures a nd exi ste nce prob lems for these t ypes of matroids are sugges ted.

Cited by 9Open access

Matroid theory

Gian‐Carlo Rota · Advances in Mathematics · 1983

Cited by 0

Matroid Bundles

L. Anderson · 1998

Combinatorial vector bundles, or matroid bundles, are a combinatorial analog to real vector bundles. Combinatorial objects called oriented matroids play the role of real vector spaces. This combinatorial analogy is remarkably strong, and has led to combinatorial results in topology and bundle-theoretic proofs in combinatorics. This paper surveys recent results on matroid bundles, and describes a canonical functor from real vector bundles to matroid bundles.

Cited by 0

Matroid Regression

Király, FJ, Theran, L · UCL Discovery (University College London) · 2014

We propose an algebraic combinatorial method for solving large sparse linear systems of equations locally - that is, a method which can compute single evaluations of the signal without computing the whole signal. The method scales only in the sparsity of the system and not in its size, and allows to provide error estimates for any solution method. At the heart of our approach is the so-called regression matroid, a combinatorial object associated to sparsity patterns, which allows to replace inversion of the large matrix with the inversion of a kernel matrix that is constant size. We show that our method provides the best linear unbiased estimator (BLUE) for this setting and the minimum variance unbiased estimator (MVUE) under Gaussian noise assumptions, and furthermore we show that the size of the kernel matrix which is to be inverted can be traded off with accuracy.

Cited by 0

Matroid Intersection

Jon Lee · Cambridge University Press eBooks · 2004

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

Sperner Matroid

Shyh-Nan Lee, Mau-Hsiang Shih · Archiv der Mathematik · 2003

Cited by 3

Rough matroid

William Zhu, Shiping Wang · 2011

This paper proposes a framework to integrate rough sets with matroids. Specifically, we propose the lower and upper rough matroids which extend the matroid. They are established by the lower and upper approximations of generalized rough sets based on relations, respectively. As a generalization of the lower (upper) rough matroid, the lower (upper) rough greedoid is defined, and it also a generalization of the greedoid. Moreover, an axiom of poset matroid is provided by generalized rough sets based on relations. Finally, we prove that the poset matroid is a special case of the lower rough greedoid.

Cited by 8

Matroid Homology

Tatiana Alekseyevskaya, Alexandre Borovik, Israel M. Gelfand et al. · Birkhäuser Boston eBooks · 2000

We construct chain complexes of matroids and Lagrangian symplectic matroids which generalize Kontsevich’s graph homology.

Cited by 0

Matroid theory

Jack E. Graver, Brigitte Servatius, Herman Servatius · Graduate studies in mathematics · 1993

Cited by 0

Matroid Connectivity.

John Leo · 1996

This dissertation has three parts. The first part, Chapter 1, considers the coefficient $b\sb{ij}(M)$ of $x\sp{i}y\sp{j}$ in the Tutte polynomial of a connected matroid M. The main result characterizes, for each i and j, the minor-minimal such matroids for which $b\sb{ij}(M)>0.$ One consequence of this characterization is that $b\sb{11}(M)>0$ if and only if the two-wheel is a minor of M. Similar results are obtained for other values of i and j. These results imply that if M is simple and representable over $GF(q),$ then there are coefficients of its Tutte polynomial which count the flats of M that are projective spaces of specified rank. Similarly, for a simple graphic matroid $M(G),$ there are coefficients that count the number of cliques of each size contained in G. The second part, Chapter 2, generalizes a graph result of Mader by proving that if f is an element of a circuit C of a 3-connected matroid M and $M\\ e$ is not 3-connected for each $e\in C-\{f\},$ then C meets a triad of M. Several consequences of this result are proved. One of these generalizes a graph result of Wu. The others provide 3-connected analogues of 2-connected matroid results of Oxley. The third part, Chapters 3-5, involves a decomposition of 3-connected binary matroids based on 3-separations and three-sums. The dual of this decomposition is a direct generalization of a decomposition due to Coullard, Gardner, and Wagner for 3-connected graphs. In Chapter 3, we define the decomposition and prove that minimal such decompositions are unique. In Chapter 4, the components of this decomposition are characterized. In Chapter 5, it is shown that, when restricted to contraction-minimally 3-connected binary matroids, the components that are not vertically 4-connected are wheels, duals of twirls, or binary spikes.

Cited by 0Open access

Matroid bundles

Laura Anderson · 1999

Combinatorial vector bundles, or matroid bundles, are a combinatorial analog to real vector bundles. Combinatorial objects called oriented matroids play the role of real vector spaces. This combinatorial analogy is remarkably strong, and has led to combinatorial results in topology and bundle-theoretic proofs in combinatorics. This paper surveys recent results on matroid bundles, and describes a canonical functor from real vector bundles to matroid bundles.

Cited by 0

Matroid Theory

James Oxley · Civil War Book Review · 2011

* What is the essence of the similarity between linearly independent sets of columns of a matrix and forests in a graph?* Why does the greedy algorithm produce a spanning tree of minimum weight in a connected graph?* Can we test in polynomial time whether a matrix is totally unimodular?Matroid theory examines and answers questions like these. Seventy-five years of study of matroids has seen the development of a rich theory with links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and theirapplications extend into electrical and structural engineering.This book falls into two parts: the first provides a comprehensive introduction to the basics of matroid theory, while the second treats more advanced topics. The book contains over seven hundred exercises and includes, for the first time in one place, proofs of all of the major theorems in thesubject. The last two chapters review current research and list more than eighty unsolved problems along with a description of the progress towards their solutions.

Cited by 0

Matroid designs

Young, P; Edmonds, J · National Institute of Standards and Technology (NIST)

Matroids are investigated in which equicardinality conditions are imposed on the flats. Such matroids are shown to be closely related to certain types of BIBD's. Existence and uniqueness theorems for these structures are derived, together with a boundedness criterion on their rank. Several classes are exhibited, including finite projective and affine geometries, certain tdesigns (Steiner systems) and so-called trivioids. By viewing certain t-designs as matroids, new ways of constructing BIBD's are derived. Three new series of 3-designs and two new 4-designs are obtained by these methods. A matroid analysis of the 5(24,8,1) design of Witt is presented, and examples are obtained from it of matroids having equicardinal hyperplanes but not equicardinal flats in lower ranks. Several general conjectures and existence problems for these types of matroids are suggested.

Cited by 0Open access