Topics in Matroid Theory

Author: Leonidas S. Pitsoulis

Publisher: Springer Science & Business Media

ISBN: 1461489571

Category: Mathematics

Page: 127

View: 7549

Topics in Matroid Theory provides a brief introduction to matroid theory with an emphasis on algorithmic consequences.Matroid theory is at the heart of combinatorial optimization and has attracted various pioneers such as Edmonds, Tutte, Cunningham and Lawler among others. Matroid theory encompasses matrices, graphs and other combinatorial entities under a common, solid algebraic framework, thereby providing the analytical tools to solve related difficult algorithmic problems. The monograph contains a rigorous axiomatic definition of matroids along with other necessary concepts such as duality, minors, connectivity and representability as demonstrated in matrices, graphs and transversals. The author also presents a deep decomposition result in matroid theory that provides a structural characterization of graphic matroids, and show how this can be extended to signed-graphic matroids, as well as the immediate algorithmic consequences.

Matroid Theory

Author: J. G. Oxley

Publisher: Oxford University Press, USA

ISBN: 9780199202508

Category: Mathematics

Page: 532

View: 5240

The study of matroids is a branch of discrete mathematics with basic links to graphs, lattices, codes, transversals, and projective geometries. Matroids are of fundamental importance in combinatorial optimization and their applications extend into electrical engineering and statics. This new in paperback version of the classic "Matroid Theory" by James Oxley provides a comprehensive introduction to matroid theory, covering the very basics to more advanced topics. With over 500 exercisesand proofs of major theorems, this book is the ideal reference and class text for academics and graduate students in mathematics and computer science. The final chapter lists sixty unsolved problems and describes progress towards their solutions.

Matroid Theory

AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory, July 2-6, 1995, University of Washington, Seattle

Author: Joseph Edmond Bonin

Publisher: American Mathematical Soc.

ISBN: 0821805088

Category: Mathematics

Page: 418

View: 5682

This volume contains the proceedings of the 1995 AMS-IMS-SIAM Joint Summer Research Conference on Matroid Theory held at the University of Washington, Seattle. The book features three comprehensive surveys that bring the reader to the forefront of research in matroid theory. Joseph Kung's encyclopedic treatment of the critical problem traces the development of this problem from its origins through its numerous links with other branches of mathematics to the current status of its many aspects. James Oxley's survey of the role of connectivity and structure theorems in matroid theory stresses the influence of the Wheels and Whirls Theorem of Tutte and the Splitter Theorem of Seymour. Walter Whiteley's article unifies applications of matroid theory to constrained geometrical systems, including the rigidity of bar-and-joint frameworks, parallel drawings, and splines. These widely accessible articles contain many new results and directions for further research and applications. The surveys are complemented by selected short research papers. The volume concludes with a chapter of open problems. Features self-contained, accessible surveys of three active research areas in matroid theory; many new results; pointers to new research topics; a chapter of open problems; mathematical applications; and applications and connections to other disciplines, such as computer-aided design and electrical and structural engineering.

Theory of Matroids

Author: Neil White

Publisher: Cambridge University Press

ISBN: 0521309379

Category: Mathematics

Page: 316

View: 1669

The theory of matroids is unique in the extent to which it connects such disparate branches of combinatorial theory and algebra as graph theory, lattice theory, design theory, combinatorial optimization, linear algebra, group theory, ring theory and field theory. Furthermore, matroid theory is alone among mathematical theories because of the number and variety of its equivalent axiom systems. Indeed, matroids are amazingly versatile and the approaches to the subject are varied and numerous. This book is a primer in the basic axioms and constructions of matroids. The contributions by various leaders in the field include chapters on axiom systems, lattices, basis exchange properties, orthogonality, graphs and networks, constructions, maps, semi-modular functions and an appendix on cryptomorphisms. The authors have concentrated on giving a lucid exposition of the individual topics; explanations of theorems are preferred to complete proofs and original work is thoroughly referenced. In addition, exercises are included for each topic.

Matroid Theory

Author: D. J. A. Welsh

Publisher: Courier Corporation

ISBN: 0486474399

Category: Mathematics

Page: 433

View: 2343

The theory of matroids connects disparate branches of combinatorial theory and algebra such as graph and lattice theory, combinatorial optimization, and linear algebra. This text describes standard examples and investigation results, and it uses elementary proofs to develop basic matroid properties before advancing to a more sophisticated treatment. 1976 edition.

Topics in Combinatorics and Graph Theory

Essays in Honour of Gerhard Ringel

Author: Rainer Bodendiek,Rudolf Henn

Publisher: Springer Science & Business Media

ISBN: 3642469086

Category: Mathematics

Page: 792

View: 6043

Graph Theory is a part of discrete mathematics characterized by the fact of an extremely rapid development during the last 10 years. The number of graph theoretical paper as well as the number of graph theorists increase very strongly. The main purpose of this book is to show the reader the variety of graph theoretical methods and the relation to combinatorics and to give him a survey on a lot of new results, special methods, and interesting informations. This book, which grew out of contributions given by about 130 authors in honour to the 70th birthday of Gerhard Ringel, one of the pioneers in graph theory, is meant to serve as a source of open problems, reference and guide to the extensive literature and as stimulant to further research on graph theory and combinatorics.

A Source Book in Matroid Theory

Author: KUNG

Publisher: Springer Science & Business Media

ISBN: 1468491997

Category: Mathematics

Page: 413

View: 7308

by Gian-Carlo Rota The subjects of mathematics, like the subjects of mankind, have finite lifespans, which the historian will record as he freezes history at one instant of time. There are the old subjects, loaded with distinctions and honors. As their problems are solved away and the applications reaped by engineers and other moneymen, ponderous treatises gather dust in library basements, awaiting the day when a generation as yet unborn will rediscover the lost paradise in awe. Then there are the middle-aged subjects. You can tell which they are by roaming the halls of Ivy League universities or the Institute for Advanced Studies. Their high priests haughtily refuse fabulous offers from eager provin cial universities while receiving special permission from the President of France to lecture in English at the College de France. Little do they know that the load of technicalities is already critical, about to crack and submerge their theorems in the dust of oblivion that once enveloped the dinosaurs. Finally, there are the young subjects-combinatorics, for instance. Wild eyed individuals gingerly pick from a mountain of intractable problems, chil dishly babbling the first words of what will soon be a new language. Child hood will end with the first Seminaire Bourbaki. It could be impossible to find a more fitting example than matroid theory of a subject now in its infancy. The telltale signs, for an unfailing diagnosis, are the abundance of deep theorems, going together with a paucity of theories.

Matroids: A Geometric Introduction

Author: Gary Gordon,Jennifer McNulty

Publisher: Cambridge University Press

ISBN: 0521145686

Category: Language Arts & Disciplines

Page: 393

View: 6581

This friendly introduction helps undergraduate students understand and appreciate matroid theory and its connections to geometry.

Matroid Applications

Author: Neil White

Publisher: Cambridge University Press

ISBN: 9780521381659

Category: Mathematics

Page: 363

View: 5439

This volume, the third in a sequence that began with The Theory of Matroids and Combinatorial Geometries, concentrates on the applications of matroid theory to a variety of topics from engineering (rigidity and scene analysis), combinatorics (graphs, lattices, codes and designs), topology and operations research (the greedy algorithm).

Combinatorics

Topics, Techniques, Algorithms

Author: Peter J. Cameron

Publisher: Cambridge University Press

ISBN: 110739337X

Category: Mathematics

Page: N.A

View: 1280

Combinatorics is a subject of increasing importance, owing to its links with computer science, statistics and algebra. This is a textbook aimed at second-year undergraduates to beginning graduates. It stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that a constructive or algorithmic proof is more valuable than an existence proof. The book is divided into two parts, the second at a higher level and with a wider range than the first. Historical notes are included which give a wider perspective on the subject. More advanced topics are given as projects and there are a number of exercises, some with solutions given.

Selected topics in discrete mathematics: Proceedings of the Moscow Discrete Mathematics Seminar, 1972-1990

Author: Alexander K. Kelmans

Publisher: American Mathematical Soc.

ISBN: 9780821895924

Category:

Page: N.A

View: 4388

This is a collection of translations of a variety of papers on discrete mathematics by members of the Moscow Seminar on Discrete Mathematics. This seminar, begun in 1972, was marked by active participation and intellectual ferment. Mathematicians in the USSR often encountered difficulties in publishing, so many interesting results in discrete mathematics remained unknown in the West for some years, and some are unknown even to the present day. To help fill this communication gap, this collection offers papers that were obscurely published and very hard to find. Among the topics covered here are: graph theory, network flow and multicommodity flow, linear programming and combinatorial optimization, matroid theory and submodular systems, matrix theory and combinatorics, parallel computing, complexity of algorithms, random graphs and statistical mechanics, coding theory, and algebraic combinatorics and group theory.

Combinatorial Optimization

Theory and Algorithms

Author: Bernhard Korte,Jens Vygen

Publisher: Springer

ISBN: 3662560399

Category: Mathematics

Page: 698

View: 9808

This comprehensive textbook on combinatorial optimization places special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. It is based on numerous courses on combinatorial optimization and specialized topics, mostly at graduate level. This book reviews the fundamentals, covers the classical topics (paths, flows, matching, matroids, NP-completeness, approximation algorithms) in detail, and proceeds to advanced and recent topics, some of which have not appeared in a textbook before. Throughout, it contains complete but concise proofs, and also provides numerous exercises and references. This sixth edition has again been updated, revised, and significantly extended. Among other additions, there are new sections on shallow-light trees, submodular function maximization, smoothed analysis of the knapsack problem, the (ln 4+ɛ)-approximation for Steiner trees, and the VPN theorem. Thus, this book continues to represent the state of the art of combinatorial optimization.

Fractional Graph Theory

A Rational Approach to the Theory of Graphs

Author: Edward R. Scheinerman,Daniel H. Ullman

Publisher: Courier Corporation

ISBN: 0486292134

Category: Mathematics

Page: 240

View: 8141

This volume explains the general theory of hypergraphs and presents in-depth coverage of fundamental and advanced topics: fractional matching, fractional coloring, fractional edge coloring, fractional arboricity via matroid methods, fractional isomorphism, and more. 1997 edition.

Matroid Decomposition

Author: K. Truemper

Publisher: Leibniz Company

ISBN: 1483276627

Category: Mathematics

Page: 408

View: 7506

Matroid Decomposition deals with decomposition and composition of matroids. The emphasis is on binary matroids, which are produced by the matrices over the binary field GF(2). Different classes of matroids are described (graphic, regular, almost regular, max-flow min-cut), along with polynomial testing algorithms. Representative applications and, except for the almost-regular case, characterizations in terms of excluded minors are given. In addition, excluded minor characterizations of both binary and ternary matroids are presented. Comprised of 13 chapters, this book begins with an introduction to basic definitions concerning graphs and matrices, followed by a discussion on binary matroids. Subsequent chapters focus on some elementary constructions of graphs and binary matroids; a simple yet effective method called the path shortening technique for establishing basic connectivity relationships and certain results about the intersection and partitioning of matroids; an algorithm for identifying certain matroid separations; and the so-called splitter theorem. Fundamental notions and theorems about matroid decomposition and composition are described, along with a very important property of real matrices called total unimodularity. The book concludes with an analysis of flows in matroids based on ideas from flows in graphs. This monograph will be of interest to students and practitioners in diverse fields such as civil, electrical, and mechanical engineering, as well as computer science and mathematics.

Coxeter Matroids

Author: Alexandre V. Borovik,Israel M. Gelfand,Neil White

Publisher: Springer Science & Business Media

ISBN: 1461220661

Category: Mathematics

Page: 266

View: 6457

Matroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry, and "Coxeter Matroids" provides an intuitive and interdisciplinary treatment of their theory. In this text, matroids are examined in terms of symmetric and finite reflection groups; also, symplectic matroids and the more general coxeter matroids are carefully developed. The Gelfand-Serganova theorem, which allows for the geometric interpretation of matroids as convex polytopes with certain symmetry properties, is presented, and in the final chapter, matroid representations and combinatorial flag varieties are discussed. With its excellent bibliography and index and ample references to current research, this work will be useful for graduate students and research mathematicians.

Combinatorial Optimization

Algorithms and Complexity

Author: Christos H. Papadimitriou,Kenneth Steiglitz

Publisher: Courier Corporation

ISBN: 0486320138

Category: Mathematics

Page: 528

View: 8081

This graduate-level text considers the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; local search heuristics for NP-complete problems, more. 1982 edition.

Combinatorial Group Theory

Presentations of Groups in Terms of Generators and Relations

Author: Wilhelm Magnus,Abraham Karrass,Donald Solitar

Publisher: Courier Corporation

ISBN: 0486438309

Category: Mathematics

Page: 444

View: 6231

This seminal, much-cited account begins with a fairly elementary exposition of basic concepts and a discussion of factor groups and subgroups. The topics of Nielsen transformations, free and amalgamated products, and commutator calculus receive detailed treatment. The concluding chapter surveys word, conjugacy, and related problems; adjunction and embedding problems; and more. Second, revised 1976 edition.

A Source Book in Matroid Theory

Author: KUNG

Publisher: Springer Science & Business Media

ISBN: 1468491997

Category: Mathematics

Page: 413

View: 8963

by Gian-Carlo Rota The subjects of mathematics, like the subjects of mankind, have finite lifespans, which the historian will record as he freezes history at one instant of time. There are the old subjects, loaded with distinctions and honors. As their problems are solved away and the applications reaped by engineers and other moneymen, ponderous treatises gather dust in library basements, awaiting the day when a generation as yet unborn will rediscover the lost paradise in awe. Then there are the middle-aged subjects. You can tell which they are by roaming the halls of Ivy League universities or the Institute for Advanced Studies. Their high priests haughtily refuse fabulous offers from eager provin cial universities while receiving special permission from the President of France to lecture in English at the College de France. Little do they know that the load of technicalities is already critical, about to crack and submerge their theorems in the dust of oblivion that once enveloped the dinosaurs. Finally, there are the young subjects-combinatorics, for instance. Wild eyed individuals gingerly pick from a mountain of intractable problems, chil dishly babbling the first words of what will soon be a new language. Child hood will end with the first Seminaire Bourbaki. It could be impossible to find a more fitting example than matroid theory of a subject now in its infancy. The telltale signs, for an unfailing diagnosis, are the abundance of deep theorems, going together with a paucity of theories.