DOCSLIB.ORG

Applications of Matroid Theory and Combinatorial Optimization to Information and Coding Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Applications of Matroid Theory and Combinatorial Optimization to Information and Coding Theory Applications of Matroid Theory and Combinatorial Optimization to Information and Coding Theory Navin Kashyap (Queen’s University), Emina Soljanin (Alcatel-Lucent Bell Labs) Pascal Vontobel (Hewlett-Packard Laboratories) August 2–7, 2009 The aim of this workshop was to bring together experts and students from pure and applied mathematics, computer science, and engineering, who are working on related problems in the areas of matroid theory, combinatorial optimization, coding theory, secret sharing, network coding, and information inequalities. The goal was to foster exchange of mathematical ideas and tools that can help tackle some of the open problems of central importance in coding theory, secret sharing, and network coding, and at the same time, to get pure mathematicians and computer scientists to be interested in the kind of problems that arise in these applied fields. 1 Introduction Matroids are structures that abstract certain fundamental properties of dependence common to graphs and vector spaces. The theory of matroids has its origins in graph theory and linear algebra, and its most successful applications in the past have been in the areas of combinatorial optimization and network theory. Recently, however, there has been a flurry of new applications of this theory in the fields of information and coding theory. It is only natural to expect matroid theory to have an influence on the theory of error-correcting codes, as matrices over finite fields are objects of fundamental importance in both these areas of mathematics. Indeed, as far back as 1976, Greene [7] (re-)derived the MacWilliams identities — which relate the Hamming weight enumerators of a linear code and its dual — as special cases of an identity for the Tutte polynomial of a matroid. However, aside from such use of tools from matroid theory to re-derive results in coding theory that had already been proved by other means, each field has had surprisingly little impact on the other, until very recently. Matroid-theoretic methods are now starting to play an important role in the understanding of decoding algorithms for error-correcting codes. In a parallel and largely unrelated development, ideas from matroid theory are also finding other novel applications within the broader realm of information theory. Specifically, they are being applied to explore the fundamental limits of secret sharing schemes and network coding, and also to gain an understanding of information inequalities. We outline some of these recent developments next. 2 Background and Recent Developments Our workshop covered four major areas within the realm of information theory — coding theory, secret sharing, network coding, and information inequalities — which have seen a recent influx of ideas from 1 2 matroid theory and combinatorial optimization. We briefly discuss the applications of such ideas in each of these areas in turn. 2.1 Coding Theory The serious study of (channel) coding theory started with Shannon’s monumental 1948 paper [9]. Shannon stated the result that reliable communication is possible at rates up to channel capacity, meaning that for any desired symbol or block error probability there exists a channel code and a decoding algorithm that can achieve this symbol or block error probability as long as the rate of the channel code is smaller than the channel capacity. On the other hand, Shannon showed that if the rate is larger than capacity, the symbol and the block error probability must be bounded away from zero. Unfortunately, the proof of the above achievability result is nonconstructive, meaning that it shows only the existence of such channel codes and decoding algorithms. Therefore, since the appearance of Shannon’s theorem, the quest has been on to find codes with practical encoding and decoding algorithms that fulfill Shannon’s promise. The codes and decoding schemes that people have come up with can broadly be classified into two classes: “traditional schemes” and “modern schemes.” In “traditional schemes,” codes were proposed that have some desirable properties like large minimum Hamming distance (a typical example of such codes being the Reed- Solomon codes). However, given a code, it was usually unclear how to decode it efficiently. Often it took quite some time until such a decoding algorithm was found (e.g., the Berlekamp-Massey decoding algorithm for Reed-Solomon codes), if at all. In “modern schemes,” the situation is reversed: given an iterative decoding algorithm like the sum-product algorithm, the question is what codes work well together with such an iterative decoding algorithm. “Modern schemes” took off with the seminal paper by Berrou, Glavieu, and Thitimajshima in 1993 on turbo coding schemes [2]. Actually, codes and decoding algorithm in the spirit of “modern schemes” were already described in the early 1960s by Gallager in his Ph.D. thesis [5]. However, these schemes were, besides the work by Zyablov, Pinkser, and Tanner in the 1970s and 1980s, largely forgotten until the mid-1990s. Only then people started to appreciate Gallager’s revolutionary approach to coding theory. Gallager proposed to define codes in terms of graphs. Such graphs are now known as Tanner graphs: they are bipartite graphs where one class of vertices corresponds to codeword symbols and where the other class of vertices corresponds to parity-checks that are imposed on the adjacent codeword symbols. Decoding is then based on repeatedly sending messages with estimates about the value of the codeword symbols along edges, and to locally process these messages at vertices in order to produce new messages that are again sent along the edges. Especially for sparse Tanner graphs the resulting decoding algorithms have very low implementation complexity. In the last fifteen years, Tanner graphs and iterative decoding algorithms have been generalized to fac- tor graphs and algorithms operating on them, and many connections to techniques in statistical mechanics, graphical models, artificial intelligence, and combinatorial optimization were uncovered. The workshop talk by Kashyap (see Section 3.1) surveyed the connection between complexity measures for graphical models for a code and the treewidth (and other width parameters) of the associated matroid. On the other hand, the workshop talks by Wainwright and Vontobel (see Section 3.1) emphasized the connections between message- passing iterative decoding of codes and certain techniques from combinatorial optimization. In particular, they discussed the linear programming decoder by Feldman, Wainwright, and Karger [4], which is a low- complexity relaxation of an integer linear programming formulation of the maximum likelihood decoder. This linear programming decoder (and its variations) has paved the way for the use of tools from combinato- rial optimization and matroids in the design and analysis of decoding algorithms. 2.2 Secret Sharing The second major application of matroid-theoretic ideas that we mention here is with respect to secret-sharing schemes. A secret-sharing scheme is a method to distribute shares of a secret value among a certain number of participants such that qualified subsets of participants (e.g., subsets of a certain size) can recover the secret from their joint shares, but unqualified subsets of participants can obtain no information whatsoever about the secret by pooling together their shares. Secret-sharing schemes were originally motivated by the problem 3 of secure storage of cryptographic keys, but have since found numerous other applications in cryptography and distributed computing. It is not difficult to show that in a secret-sharing scheme, the size of each of the shares cannot be smaller than the size (information content) of the secret value. An ideal secret-sharing scheme is one in which all shares have the same size as the secret value. More generally, the information rate of a secret-sharing scheme is the ratio of the size of the secret to the maximum share size. In a secret-sharing scheme, the collection of qualified subsets of participants is called the access structure of the scheme. It is known that for any monotone increasing collection, Γ, of subsets of a finite set, one can define a secret-sharing scheme with access structure Γ. The information rate ρ(Γ) is defined to be the supremum of information rates among all secret-sharing schemes having access structure Γ. Γ is said to be an ideal access structure if it admits an ideal secret-sharing scheme. Brickell and Davenport [3] began a line of work relating ideal secret-sharing schemes to matroids. They showed that any ideal access structure is induced by a matroid in a very specific sense. However, it is also known that not every matroid gives rise to an ideal access structure; for example, the access structures induced by the Vamos´ matroid are not ideal. Characterizing the matroids that give rise to ideal access structures has remained an open problem. There has been some very recent work on computing the information rates of non-ideal access structures using polymatroid techniques, linear programming, and non-Shannon information inequalities. For example, it has been shown that for any access structure Γ induced by the Vamos´ matroid, ρ(Γ) ≤ 19/21, which shows that such access structures are far from being ideal. This, and other related results, were surveyed in the workshop talks by Padro´ and Beimel (see Section 3.3). Secret-sharing schemes have also been received some recent attention in the quantum domain, a topic covered in the workshop talk by Sarvepalli (see Section 3.3). 2.3 Network Coding Another novel application of matroid theory and combinatorial optimization within the realm of information theory is in the area of network coding [10]. Network coding is an elegant technique introduced at the turn of the millennium to improve network throughput and performance. Since then, it has attracted significant interest from diverse scientific communities of engineers, computer scientists, and mathematicians in both academia and industry. This workshop explored connections between network coding and combinatorial optimization, matroids, and non-Shannon inequalities.
Load more

Recommended publications
  • Matroid Theory
    MATROID THEORY HAYLEY HILLMAN 1 2 HAYLEY HILLMAN Contents 1. Introduction to Matroids 3 1.1. Basic Graph Theory 3 1.2. Basic Linear Algebra 4 2. Bases 5 2.1. An Example in Linear Algebra 6 2.2. An Example in Graph Theory 6 3. Rank Function 8 3.1. The Rank Function in Graph Theory 9 3.2. The Rank Function in Linear Algebra 11 4. Independent Sets 14 4.1. Independent Sets in Graph Theory 14 4.2. Independent Sets in Linear Algebra 17 5. Cycles 21 5.1. Cycles in Graph Theory 22 5.2. Cycles in Linear Algebra 24 6. Vertex-Edge Incidence Matrix 25 References 27 MATROID THEORY 3 1. Introduction to Matroids A matroid is a structure that generalizes the properties of indepen- dence. Relevant applications are found in graph theory and linear algebra. There are several ways to define a matroid, each relate to the concept of independence. This paper will focus on the the definitions of a matroid in terms of bases, the rank function, independent sets and cycles. Throughout this paper, we observe how both graphs and matrices can be viewed as matroids. Then we translate graph theory to linear algebra, and vice versa, using the language of matroids to facilitate our discussion. Many proofs for the properties of each definition of a matroid have been omitted from this paper, but you may find complete proofs in Oxley[2], Whitney[3], and Wilson[4]. The four definitions of a matroid introduced in this paper are equiv- alent to each other.
    [Show full text]
  • Matroids and Tropical Geometry
    matroids unimodality and log concavity strategy: geometric models tropical models directions Matroids and Tropical Geometry Federico Ardila San Francisco State University (San Francisco, California) Mathematical Sciences Research Institute (Berkeley, California) Universidad de Los Andes (Bogotá, Colombia) Introductory Workshop: Geometric and Topological Combinatorics MSRI, September 5, 2017 (3,-2,0) Who is here? • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • Who is here? • matroids unimodality and log concavity strategy: geometric models tropical models directions Preface. Thank you, organizers! • Who is here? • This is the Introductory Workshop. • Focus on accessibility for grad students and junior faculty. • # (questions by students + postdocs) # (questions by others) • ≥ matroids unimodality and log concavity strategy: geometric models tropical models directions Summary. Matroids are everywhere. • Many matroid sequences are (conj.) unimodal, log-concave. • Geometry helps matroids. • Tropical geometry helps matroids and needs matroids. • (If time) Some new constructions and results. • Joint with Carly Klivans (06), Graham Denham+June Huh (17). Properties: (B1) B = /0 6 (B2) If A;B B and a A B, 2 2 − then there exists b B A 2 − such that (A a) b B. − [ 2 Definition. A set E and a collection B of subsets of E are a matroid if they satisfies properties (B1) and (B2).
    [Show full text]
  • Matroids You Have Known
    26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction.
    [Show full text]
  • On Decomposing a Hypergraph Into K Connected Sub-Hypergraphs
    Egervary´ Research Group on Combinatorial Optimization Technical reportS TR-2001-02. Published by the Egrerv´aryResearch Group, P´azm´any P. s´et´any 1/C, H{1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres . ISSN 1587{4451. On decomposing a hypergraph into k connected sub-hypergraphs Andr´asFrank, Tam´asKir´aly,and Matthias Kriesell February 2001 Revised July 2001 EGRES Technical Report No. 2001-02 1 On decomposing a hypergraph into k connected sub-hypergraphs Andr´asFrank?, Tam´asKir´aly??, and Matthias Kriesell??? Abstract By applying the matroid partition theorem of J. Edmonds [1] to a hyper- graphic generalization of graphic matroids, due to M. Lorea [3], we obtain a gen- eralization of Tutte's disjoint trees theorem for hypergraphs. As a corollary, we prove for positive integers k and q that every (kq)-edge-connected hypergraph of rank q can be decomposed into k connected sub-hypergraphs, a well-known result for q = 2. Another by-product is a connectivity-type sufficient condition for the existence of k edge-disjoint Steiner trees in a bipartite graph. Keywords: Hypergraph; Matroid; Steiner tree 1 Introduction An undirected graph G = (V; E) is called connected if there is an edge connecting X and V X for every nonempty, proper subset X of V . Connectivity of a graph is equivalent− to the existence of a spanning tree. As a connected graph on t nodes contains at least t 1 edges, one has the following alternative characterization of connectivity. − Proposition 1.1. A graph G = (V; E) is connected if and only if the number of edges connecting distinct classes of is at least t 1 for every partition := V1;V2;:::;Vt of V into non-empty subsets.P − P f g ?Department of Operations Research, E¨otv¨osUniversity, Kecskem´etiu.
    [Show full text]
  • Matroid Bundles
    New Perspectives in Geometric Combinatorics MSRI Publications Volume 38, 1999 Matroid Bundles LAURA ANDERSON Abstract. Combinatorial vector bundles, or matroid bundles,areacom- binatorial analog to real vector bundles. Combinatorial objects called ori- ented matroids play the role of real vector spaces. This combinatorial anal- ogy 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. 1. Introduction Matroid bundles are combinatorial objects that mimic real vector bundles. They were first defined in [MacPherson 1993] in connection with combinatorial differential manifolds,orCD manifolds. Matroid bundles generalize the notion of the “combinatorial tangent bundle” of a CD manifold. Since the appearance of McPherson’s article, the theory has filled out considerably; in particular, ma- troid bundles have proved to provide a beautiful combinatorial formulation for characteristic classes. We will recapitulate many of the ideas introduced by McPherson, both for the sake of a self-contained exposition and to describe them in terms more suited to our present context. However, we refer the reader to [MacPherson 1993] for background not given here. We recommend the same paper, as well as [Mn¨ev and Ziegler 1993] on the combinatorial Grassmannian, for related discussions. We begin with a key intuitive point of the theory: the notion of an oriented matroid as a combinatorial analog to a vector space. From this we develop matroid bundles as a combinatorial bundle theory with oriented matroids as fibers. Section 2 will describe the category of matroid bundles and its relation to the category of real vector bundles.
    [Show full text]
  • SHIFTED MATROID COMPLEXES 1. Introduction We Want to Consider
    SHIFTED MATROID COMPLEXES CAROLINE J. KLIVANS Abstract. We study the class of matroids whose independent set complexes are shifted simplicial complexes. We prove two characterization theorems, one of which is constructive. In addition, we show this class is closed under taking minors and duality. Finally, we give results on shifted broken circuit complexes. 1. Introduction We want to consider those matroids whose complexes of independent sets are shifted. This class of shifted matroids has been significantly investigated before, in a variety of guises. They seem to have been first used in [5] by Crapo. He used them to show that there are at least 2n non-isomorphic matroids on n elements. They were later investigated in [11] as a particular kind of transversal matroid. This work includes a characterization in terms of forbidden minors. Shifted matroids appeared again in [2] and [3] as a kind of lattice path matroid. Mostly closely related to the presentation here is [1], where these matroids arise in the context of Catalan matroids. We refer the reader to [3] for a more complete history of these matroids. Our approach is to consider the matroid complexes specifically from a shifted perspective. We provide new proofs utilizing the structure of a shifted complex to show the class is constructible, closed under taking minors and duality, and exactly the set of principal order ideals in the shifted partial ordering. 2. Shifted complexes A simplicial complex on n vertices is shifted if there exists a labeling of the vertices by one through n such that for any face fv1; v2; : : : ; vkg, replacing any vi by a vertex with a smaller label results in a collection which is also a face.
    [Show full text]
  • Topics in Combinatorial Optimization
    Topics in Combinatorial Optimization Orlando Lee – Unicamp 4 de junho de 2014 Orlando Lee – Unicamp Topics in Combinatorial Optimization Agradecimentos Este conjunto de slides foram preparados originalmente para o curso T´opicos de Otimiza¸c˜ao Combinat´oria no primeiro semestre de 2014 no Instituto de Computa¸c˜ao da Unicamp. Preparei os slides em inglˆes simplesmente porque me deu vontade, mas as aulas ser˜ao em portuguˆes (do Brasil)! Agradecimentos especiais ao Prof. M´ario Leston Rey. Sem sua ajuda, certamente estes slides nunca ficariam prontos a tempo. Qualquer erro encontrado nestes slide ´ede minha inteira responsabilidade (Orlando Lee, 2014). Orlando Lee – Unicamp Topics in Combinatorial Optimization Independent sets A matroid is a pair M = (E, I) in which I ⊆ 2E that satisfies the following properties: (I1) ∅∈ I. (I2) If I ∈ I and I ′ ⊆ I , then I ′ ∈ I. (I3) If I 1, I 2 ∈ I and |I 1| < |I 2|, then there exists e ∈ I 2 − I 1 such that I 1 ∪ {e}∈ I. (Independence augmenting axiom) We say that the members of I are the independent sets of M. Orlando Lee – Unicamp Topics in Combinatorial Optimization Matroid intersection We have seen how to determine efficiently (via an independence oracle) a maximum basis (independent set) of a matroid. Let M1 = (E, I1) and M2 = (E , I2) be two matroids. We are interested in finding a maximum element I ∈ I1 ∩ I2, that is, a maximum common independent set I of both matroids. We consider the unweighted version, in which we want to maximize |I | for I ∈ I1 ∩ I2, and the weighted version in which given c : E 7→ R we want to maximum c(I ) for I ∈ I1 ∩ I2.
    [Show full text]
  • Matroid Matching: the Power of Local Search
    Matroid Matching: the Power of Local Search [Extended Abstract] Jon Lee Maxim Sviridenko Jan Vondrák IBM Research IBM Research IBM Research Yorktown Heights, NY Yorktown Heights, NY San Jose, CA [email protected] [email protected] [email protected] ABSTRACT can be easily transformed into an NP-completeness proof We consider the classical matroid matching problem. Un- for a concrete class of matroids (see [46]). An important weighted matroid matching for linear matroids was solved result of Lov´asz is that (unweighted) matroid matching can by Lov´asz, and the problem is known to be intractable for be solved in polynomial time for linear matroids (see [35]). general matroids. We present a PTAS for unweighted ma- There have been several attempts to generalize Lov´asz' re- troid matching for general matroids. In contrast, we show sult to the weighted case. Polynomial-time algorithms are that natural LP relaxations have an Ω(n) integrality gap known for some special cases (see [49]), but for general linear and moreover, Ω(n) rounds of the Sherali-Adams hierarchy matroids there is only a pseudopolynomial-time randomized are necessary to bring the gap down to a constant. exact algorithm (see [8, 40]). More generally, for any fixed k ≥ 2 and > 0, we obtain a In this paper, we revisit the matroid matching problem (k=2 + )-approximation for matroid matching in k-uniform for general matroids. Our main result is that while LP- hypergraphs, also known as the matroid k-parity problem. based approaches including the Sherali-Adams hierarchy fail As a consequence, we obtain a (k=2 + )-approximation for to provide any meaningful approximation, a simple local- the problem of finding the maximum-cardinality set in the search algorithm gives a PTAS (in the unweighted case).
    [Show full text]
  • Homework #4 CS675 Fall 2019
    Homework #4 CS675 Fall 2019 Due Monday Dec 2 by 4:10pm General Instructions The following assignment is meant to be challenging. Feel free to discuss with fellow students, though please write up your solutions independently and acknowledge everyone you discussed the homework with on your writeup. I also expect that you will not attempt to consult outside sources, on the Internet or otherwise, for solutions to any of these homework problems. Finally, please provide a formal mathematical proof for all your claims. Problem 1. (9 points) n Let X = fv1; : : : ; vmg where vi 2 R for i = 1; : : : ; m. As explained in class, the set system over X whose feasible sets are the linearly independent subsets of X is a matroid. In this problem, we will explore whether other forms of independence also yield matroids. (a) [3 points]. Suppose instead that we define S ⊆ X to be feasible if the vectors in S are independent with respect to convex combinations | i.e. no vector v 2 S can be written as a convex combination of the remaining vectors S n v. Is the resulting set system a matroid? Prove or disprove. (b) [3 points]. What if we define S ⊆ X to be feasible if the vectors in S are affinely independent | i.e. no vector v 2 S can be written as an affine combination of the remaining vectors S n v. Is the resulting set system a matroid? Prove or disprove. (c) [3 points]. Finally, what if we say S ⊆ X is feasible if the vectors in S are conically independent | i.e.
    [Show full text]
  • 5. Matroid Optimization 5.1 Definition of a Matroid
    Massachusetts Institute of Technology 18.453: Combinatorial Optimization Michel X. Goemans April 20, 2017 5. Matroid optimization 5.1 Definition of a Matroid Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is denoted by I or I(M), and we typically refer to a matroid M by listing its ground set and its family of independent sets: M = (E; I). For M to be a matroid, I must satisfy two main axioms: (I1) if X ⊆ Y and Y 2 I then X 2 I, (I2) if X 2 I and Y 2 I and jY j > jXj then 9e 2 Y n X : X [ feg 2 I. In words, the second axiom says that if X is independent and there exists a larger independent set Y then X can be extended to a larger independent by adding an element of Y nX. Axiom (I2) implies that every maximal (inclusion-wise) independent set is maximum; in other words, all maximal independent sets have the same cardinality. A maximal independent set is called a base of the matroid. Examples. • One trivial example of a matroid M = (E; I) is a uniform matroid in which I = fX ⊆ E : jXj ≤ kg; for a given k.
    [Show full text]
  • 4. Lecture Notes on Matroid Optimization 4.1 Definition of A
    Massachusetts Institute of Technology Handout 9 18.433: Combinatorial Optimization March 20th, 2009 Michel X. Goemans 4. Lecture notes on matroid optimization 4.1 Definition of a Matroid Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is denoted by I or I(M), and we typically refer to a matroid M by listing its ground set and its family of independent sets: M = (E; I). For M to be a matroid, I must satisfy two main axioms: (I1) if X ⊆ Y and Y 2 I then X 2 I, (I2) if X 2 I and Y 2 I and jY j > jXj then 9e 2 Y n X : X [ feg 2 I. In words, the second axiom says that if X is independent and there exists a larger independent set Y then X can be extended to a larger independent by adding an element of Y nX. Axiom (I2) implies that every maximal (inclusion-wise) independent set is maximum; in other words, all maximal independent sets have the same cardinality. A maximal independent set is called a base of the matroid. Examples. • One trivial example of a matroid M = (E; I) is a uniform matroid in which I = fX ⊆ E : jXj ≤ kg; for a given k.
    [Show full text]
  • On the Graphic Matroid Parity Problem
    On the graphic matroid parity problem Zolt´an Szigeti∗ Abstract A relatively simple proof is presented for the min-max theorem of Lov´asz on the graphic matroid parity problem. 1 Introduction The graph matching problem and the matroid intersection problem are two well-solved problems in Combi- natorial Theory in the sense of min-max theorems [2], [3] and polynomial algorithms [4], [3] for finding an optimal solution. The matroid parity problem, a common generalization of them, turned out to be much more difficult. For the general problem there does not exist polynomial algorithm [6], [8]. Moreover, it contains NP-complete problems. On the other hand, for linear matroids Lov´asz provided a min-max formula in [7] and a polynomial algorithm in [8]. There are several earlier results which can be derived from Lov´asz’ theorem, e.g. Tutte’s result on f-factors [15], a result of Mader on openly disjoint A-paths [11] (see [9]), a result of Nebesky concerning maximum genus of graphs [12] (see [5]), and the problem of Lov´asz on cacti [9]. This latter one is a special case of the graphic matroid parity problem. Our aim is to provide a simple proof for the min-max formula on this problem, i. e. on the graphic matroid parity problem. In an earlier paper [14] of the present author the special case of cacti was considered. We remark that we shall apply the matroid intersection theorem of Edmonds [4]. We refer the reader to [13] for basic concepts on matroids. For a given graph G, the cycle matroid G is defined on the edge set of G in such a way that the independent sets are exactly the edge sets of the forests of G.
    [Show full text]