Matroid Theory
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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). -
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. -
Incremental Dynamic Construction of Layered Polytree Networks )
440 Incremental Dynamic Construction of Layered Polytree Networks Keung-Chi N g Tod S. Levitt lET Inc., lET Inc., 14 Research Way, Suite 3 14 Research Way, Suite 3 E. Setauket, NY 11733 E. Setauket , NY 11733 [email protected] [email protected] Abstract Many exact probabilistic inference algorithms1 have been developed and refined. Among the earlier meth ods, the polytree algorithm (Kim and Pearl 1983, Certain classes of problems, including per Pearl 1986) can efficiently update singly connected ceptual data understanding, robotics, discov belief networks (or polytrees) , however, it does not ery, and learning, can be represented as incre work ( without modification) on multiply connected mental, dynamically constructed belief net networks. In a singly connected belief network, there is works. These automatically constructed net at most one path (in the undirected sense) between any works can be dynamically extended and mod pair of nodes; in a multiply connected belief network, ified as evidence of new individuals becomes on the other hand, there is at least one pair of nodes available. The main result of this paper is the that has more than one path between them. Despite incremental extension of the singly connect the fact that the general updating problem for mul ed polytree network in such a way that the tiply connected networks is NP-hard (Cooper 1990), network retains its singly connected polytree many propagation algorithms have been applied suc structure after the changes. The algorithm cessfully to multiply connected networks, especially on is deterministic and is guaranteed to have networks that are sparsely connected. These methods a complexity of single node addition that is include clustering (also known as node aggregation) at most of order proportional to the number (Chang and Fung 1989, Pearl 1988), node elimination of nodes (or size) of the network. -
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. -
A Framework for the Evaluation and Management of Network Centrality ∗
A Framework for the Evaluation and Management of Network Centrality ∗ Vatche Ishakiany D´oraErd}osz Evimaria Terzix Azer Bestavros{ Abstract total number of shortest paths that go through it. Ever Network-analysis literature is rich in node-centrality mea- since, researchers have proposed different measures of sures that quantify the centrality of a node as a function centrality, as well as algorithms for computing them [1, of the (shortest) paths of the network that go through it. 2, 4, 10, 11]. The common characteristic of these Existing work focuses on defining instances of such mea- measures is that they quantify a node's centrality by sures and designing algorithms for the specific combinato- computing the number (or the fraction) of (shortest) rial problems that arise for each instance. In this work, we paths that go through that node. For example, in a propose a unifying definition of centrality that subsumes all network where packets propagate through nodes, a node path-counting based centrality definitions: e.g., stress, be- with high centrality is one that \sees" (and potentially tweenness or paths centrality. We also define a generic algo- controls) most of the traffic. rithm for computing this generalized centrality measure for In many applications, the centrality of a single node every node and every group of nodes in the network. Next, is not as important as the centrality of a group of we define two optimization problems: k-Group Central- nodes. For example, in a network of lobbyists, one ity Maximization and k-Edge Centrality Boosting. might want to measure the combined centrality of a In the former, the task is to identify the subset of k nodes particular group of lobbyists. -
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. -
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. -
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. -
1 Plane and Planar Graphs Definition 1 a Graph G(V,E) Is Called Plane If
1 Plane and Planar Graphs Definition 1 A graph G(V,E) is called plane if • V is a set of points in the plane; • E is a set of curves in the plane such that 1. every curve contains at most two vertices and these vertices are the ends of the curve; 2. the intersection of every two curves is either empty, or one, or two vertices of the graph. Definition 2 A graph is called planar, if it is isomorphic to a plane graph. The plane graph which is isomorphic to a given planar graph G is said to be embedded in the plane. A plane graph isomorphic to G is called its drawing. 1 9 2 4 2 1 9 8 3 3 6 8 7 4 5 6 5 7 G is a planar graph H is a plane graph isomorphic to G 1 The adjacency list of graph F . Is it planar? 1 4 56 8 911 2 9 7 6 103 3 7 11 8 2 4 1 5 9 12 5 1 12 4 6 1 2 8 10 12 7 2 3 9 11 8 1 11 36 10 9 7 4 12 12 10 2 6 8 11 1387 12 9546 What happens if we add edge (1,12)? Or edge (7,4)? 2 Definition 3 A set U in a plane is called open, if for every x ∈ U, all points within some distance r from x belong to U. A region is an open set U which contains polygonal u,v-curve for every two points u,v ∈ U. -
On Hamiltonian Line-Graphsq
ON HAMILTONIANLINE-GRAPHSQ BY GARY CHARTRAND Introduction. The line-graph L(G) of a nonempty graph G is the graph whose point set can be put in one-to-one correspondence with the line set of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of G are adjacent. In this paper graphs whose line-graphs are eulerian or hamiltonian are investigated and characterizations of these graphs are given. Furthermore, neces- sary and sufficient conditions are presented for iterated line-graphs to be eulerian or hamiltonian. It is shown that for any connected graph G which is not a path, there exists an iterated line-graph of G which is hamiltonian. Some elementary results on line-graphs. In the course of the article, it will be necessary to refer to several basic facts concerning line-graphs. In this section these results are presented. All the proofs are straightforward and are therefore omitted. In addition a few definitions are given. If x is a Une of a graph G joining the points u and v, written x=uv, then we define the degree of x by deg zz+deg v—2. We note that if w ' the point of L(G) which corresponds to the line x, then the degree of w in L(G) equals the degree of x in G. A point or line is called odd or even depending on whether it has odd or even degree. If G is a connected graph having at least one line, then L(G) is also a connected graph. -
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. -
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).