A Combinatorial Abstraction of the Shortest Path Problem and Its Relationship to Greedoids
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Arxiv:1403.0920V3 [Math.CO] 1 Mar 2019
Matroids, delta-matroids and embedded graphs Carolyn Chuna, Iain Moffattb, Steven D. Noblec,, Ralf Rueckriemend,1 aMathematics Department, United States Naval Academy, Chauvenet Hall, 572C Holloway Road, Annapolis, Maryland 21402-5002, United States of America bDepartment of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, United Kingdom cDepartment of Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom d Aschaffenburger Strasse 23, 10779, Berlin Abstract Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illus- trate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the liter- ature, including the Las Vergnas, Bollab´as-Riordanand Krushkal polynomials, are in fact delta-matroidal. Keywords: matroid, delta-matroid, ribbon graph, quasi-tree, partial dual, topological graph polynomial 2010 MSC: 05B35, 05C10, 05C31, 05C83 1. Overview Matroid theory is often thought of as a generalization of graph theory. Many results in graph theory turn out to be special cases of results in matroid theory. This is beneficial -
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. -
Polymatroid Greedoids” BERNHARD KORTE
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector IOURNAL OF COMBINATORIAL THEORY. Series B 38. 41-72 (1985) Polymatroid Greedoids” BERNHARD KORTE Bonn, West German? LASZL6 LOVASZ Budapest, Hungary Communicated by the Managing Editors Received October 5. 1983 This paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far. are demonstrated. (~‘1 1985 Acadenw Press, Inc. 1. INTRODUCTION Matroids on a finite ground set E can be defined by a rank function r:2E+n+, which is subcardinal, monotone, and submodular. Polymatroids are relaxations of them. The rank function of a polymatroid is monotone and submodular. Subcardinality is required only for the empty set by normalizing r(0) = 0. One way of defining greedoids is also via a rank function. Here, the rank function is subcardinal, monotone, but only locally submodular, which means that if the rank of a set is not increased by adding one element as well as another element to it, then the rank of the set will not increase by adding both elements to it. Again, this is a direct relaxation of submodularity and hence of the matroid definition. -
Branch-Depth: Generalizing Tree-Depth of Graphs
Branch-depth: Generalizing tree-depth of graphs ∗1 †‡23 34 Matt DeVos , O-joung Kwon , and Sang-il Oum† 1Department of Mathematics, Simon Fraser University, Burnaby, Canada 2Department of Mathematics, Incheon National University, Incheon, Korea 3Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Korea 4Department of Mathematical Sciences, KAIST, Daejeon, Korea [email protected], [email protected], [email protected] November 5, 2020 Abstract We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the no- tions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E A. For a subset X of V , \ let ρG(X) be the rank of the adjacency matrix between X and V X over the binary field. We prove that a class of graphs has bounded\ tree-depth if and only if the corresponding class of functions λG has arXiv:1903.11988v2 [math.CO] 4 Nov 2020 bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree- depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction. -
Sparsity Counts in Group-Labeled Graphs and Rigidity
Sparsity Counts in Group-labeled Graphs and Rigidity Rintaro Ikeshita1 Shin-ichi Tanigawa2 1University of Tokyo 2CWI & Kyoto University July 15, 2015 1 / 23 I Examples I k = ` = 1: forest I k = 1; ` = 0: pseudoforest I k = `: Decomposability into edge-disjoint k forests (Nash-Williams) I ` ≤ k: Decomposability into edge-disjoint k − ` pseudoforests and ` forests I general k; `: Rigidity of graphs and scene analysis (k; `)-sparsity def I A finite undirected graph G = (V ; E) is (k; `)-sparse , jF j ≤ kjV (F )j − ` for every F ⊆ E with kjV (F )j − ` ≥ 1. 2 / 23 (k; `)-sparsity def I A finite undirected graph G = (V ; E) is (k; `)-sparse , jF j ≤ kjV (F )j − ` for every F ⊆ E with kjV (F )j − ` ≥ 1. I Examples I k = ` = 1: forest I k = 1; ` = 0: pseudoforest I k = `: Decomposability into edge-disjoint k forests (Nash-Williams) I ` ≤ k: Decomposability into edge-disjoint k − ` pseudoforests and ` forests I general k; `: Rigidity of graphs and scene analysis 2 / 23 I Examples I k = ` = 1: graphic matroid I k = 1; ` = 0: bicircular matroid I ` ≤ k: union of k − ` copies of bicircular matroid and ` copies of graphic matroid I k = 2; ` = 3: generic 2-rigidity matroid (Laman70) Count Matroids I Suppose ` ≤ 2k − 1. Then Mk;`(G) = (E; Ik;`) forms a matroid, called the (k; `)-count matroid, where Ik;` = fI ⊆ E : I is (k; `)-sparseg: 3 / 23 Count Matroids I Suppose ` ≤ 2k − 1. Then Mk;`(G) = (E; Ik;`) forms a matroid, called the (k; `)-count matroid, where Ik;` = fI ⊆ E : I is (k; `)-sparseg: I Examples I k = ` = 1: graphic matroid I k = 1; ` = 0: bicircular matroid I ` ≤ k: union of k − ` copies of bicircular matroid and ` copies of graphic matroid I k = 2; ` = 3: generic 2-rigidity matroid (Laman70) 3 / 23 Group-labeled Graphs I A group-labeled graph (Γ-labeled graph) (G; ) is a directed finite graph whose edges are labeled invertibly from a group Γ. -
The Greedy Algorithm and the Cohen-Macaulay Property of Rings, Graphs and Toric Projective Curves
The Greedy Algorithm and the Cohen-Macaulay property of rings, graphs and toric projective curves Argimiro Arratia Abstract It is shown in this paper how a solution for a combinatorial problem ob- tained from applying the greedy algorithm is guaranteed to be optimal for those instances of the problem that, under an appropriate algebraic representation, satisfy the Cohen-Macaulay property known for rings and modules in Commutative Alge- bra. The choice of representation for the instances of a given combinatorial problem is fundamental for recognizing the Cohen-Macaulay property. Departing from an exposition of the general framework of simplicial complexes and their associated Stanley-Reisner ideals, wherein the Cohen-Macaulay property is formally defined, a review of other equivalent frameworks more suitable for graphs or arithmetical problems will follow. In the case of graph problems a better framework to use is the edge ideal of Rafael Villarreal. For arithmetic problems it is appropriate to work within the semigroup viewpoint of toric geometry developed by Antonio Campillo and collaborators. 1 Introduction A greedy algorithm is one of the simplest strategies to solve an optimization prob- lem. It is based on a step-by-step selection of a candidate solution that seems best at the moment, that is a local optimal solution, in the hope that in the end this process leads to a global optimal solution. Greedy algorithms do not always output (global) optimal solutions, but for many optimization problems they do. For Antonio Campillo and Miguel Angel Revilla. Argimiro Arratia Dept. Computer Science & Barcelona Graduate School of Mathematics, Universitat Politecnica` de Catalunya, Spain, e-mail: [email protected] To appear in the Festshrift volume for Antonio Campillo on the occasion of his 65th birthday (Springer 2018). -
Problem Set 2 Out: April 15 Due: April 22
CS 38 Introduction to Algorithms Spring 2014 Problem Set 2 Out: April 15 Due: April 22 Reminder: you are encouraged to work in groups of two or three; however you must turn in your own write-up and note with whom you worked. You may consult the course notes and the optional text (CLRS). The full honor code guidelines can be found in the course syllabus. Please attempt all problems. To facilitate grading, please turn in each problem on a separate sheet of paper and put your name on each sheet. Do not staple the separate sheets. 1. Recall that a prefix-free encoding scheme can be represented by a binary tree, and that the Huffman code algorithm gives an efficient way to construct an optimal such tree from the probabilities p1; p2; : : : ; pn of n symbols. In this problem, you will show that the average length of such an encoding scheme is at most one larger than the entropy (which is the information- theoretic best-possible). The entropy of the distribution given by p = (p1; p2; : : : ; pn) is defined to be Xn H(p) = − log(pi) · pi: i=1 (a) Prove that for any list of positive integers `1; `2; : : : ; `n satisfying Xn − 2 `i ≤ 1; i=1 there is a binary tree with distinct root-leaf paths having lengths `1; `2; : : : ; `n. Hint: start from the full binary tree and delete subtrees. (b) Let p = (p1; : : : ; pn) give the probabilities of n symbols. Prove that if `1; : : : ; `n are the encoding lengths in an optimal prefix-free encoding scheme for this distribution, then the average encoding length, Xn `ipi; i=1 is at most H(p) + 1. -
Efficient Algorithms for Graphic Matroid Intersection and Parity
\ 194 - 1 \ Automata, Languages and ~rogrammlng 1 , verifying Concurrent Processes Using 12th Colloquium 18 pages.1982, Nafplion, Greece, July 15-19,1985 iomatisingthe ~ogicof Computer Program- 182. s.Proceedings, 1881. Edited by 0. Kozen. ;iw Techniques I- Requirement6 and Logical 3, 1978, Edited by S.B.Yao, S.B. Navathe, .nit. V. 227 pages, 1982. Design Techniques 11: Proceedings, 1979. T.L.Kunii.V, 229-399 pages. 1992. cificalion. Proceedings, 1981. Edited by J. 366.1982. X2ProgiammingLogic.X,292 pages.108 ~age8.1982. Edited by Wilfried Brauer I on Automated Deduciion, Proceedings veland.VII. 389 pages. 1982. sopoulou. G. Per8ch.G. Goos, M. Daismann rchgassner, An Attribute Grammar lor the ida. IX, 511 pages. 1982. guages and programming.Edited by M. Niel- VII, 614 pages, 1982. 3. Hun, â Zimmermann. GAG: A Practical ', 156 pages. 1982. Springer-Verlag Berlin Heidelberg New York Tokyo Efficient Algorithms for Graphic Matroid Intersection and Parity s (Extended Abstract) algori by shorte impro Harold N.Gabow ' Matthias Stallmanu s Department of Computer Science Department of Computer Science O(n 1 University of Colorado North Carolina State University and c Boulder, CO 80309 Raleigh, NC 27695-8206 for s USA USA well- A Abstract matrc An algorithm for matmid intersection, baaed on the phase approach of Dinic for \ network flow and Hopcmft and Karp for matching, is presented. An implementation for the b graphic matmids uses time O(n1P m) if m is Of# fg n), and similar expressions indep otherwise. An implementation to find k edge-disjoint spanning trees on a graph uses eleme time O(VP nlP m) if m is 0(n lg n) and a similar expression otherwise; when m is main 0(fD I@) this improves the previous bound, 0(t? p2). -
Combinatorial Characterisations of Graphical Computational Search
Combinatorial Characterisations of accessible Graphical Computational Search Problems: P= 6 NP. Kayibi Koko-Kalambay School of Mathematics. Department of Sciences University of Bristol Abstract Let G be a connected graph with edge-set E and vertex-set V . Given an input X, where X is the edge-set or the vertex-set of the graph G, a graphical computational search problem associated with a predicate γ con- sists of finding a solution Y , where Y ⊆ X, and Y satisfies the condition γ. We denote such a problem as Π(X; γ), and we let Π^ to denote the decision problem associated with Π(X; γ). A sub-solution of Π(X; γ) is a subset Y 0 that is a solution of the problem Π(X0; γ), where X0 ⊂ X, and X0 is the edge-set or the vertex-set of the contraction-minor G=A, where A ⊆ E. To each search problem Π(X; γ), we associate the set system (X; I), where I denotes the set of all the solutions and sub-solutions of Π(X; γ). The predicate γ is an accessible predicate if, given Y 6= ;, Y satisfies γ in X implies that there is an element y 2 Y such that Y n y satisfies γ in X0. If γ is an accessible property, we then show in Theorem 1 that a decision problem Π^ is in P if and only if, for all input X, the set system (X; I) satisfies Axioms M2', where M2', called the Augmentability Axiom, is an extension of both the Exchange Axiom and the Accessibility Axiom of a greedoid. -
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. -
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. -
Arxiv:2004.00683V2 [Math.CO] 22 Jul 2020 of Symmetries: Translation-Invariance, Sn-Invariance, and Duality
UNIVERSAL TUTTE POLYNOMIAL OLIVIER BERNARDI, TAMAS´ KALM´ AN,´ ALEXANDER POSTNIKOV Abstract. The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polyno- mial. Our definition is related to previous works of Cameron and Fink and of K´alm´anand Postnikov. We then define the universal Tutte polynomial n Tn, which is a polynomial of degree n in 2 + (2 − 1) variables that special- izes to the Tutte polynomials of all polymatroids (hence all matroids) on a ground set with n elements. The universal polynomial Tn admits three kinds of symmetries: translation invariance, Sn-invariance, and duality. 1. Introduction The Tutte polynomial TM (x; y) is an important invariant of a matroid M. For a graphical matroid MG associated to a graph G, the Tutte polynomial TG(x; y) := TMG (x; y) specializes to many classical graph invariants, such as the chromatic poly- nomial, the flow polynomial, the reliability polynomial etc. The Tutte polynomial and its various evaluations were studied in the context of statistical physics (par- tition functions of the Ising and Potts models), knot theory (Jones and Kauffman polynomials), and many other areas of mathematics and physics. There is a vast literature on the Tutte polynomial; see for instance [Bol, Chapter 10] or [EM] for an introduction and references. This paper contains two main contributions to the theory of the Tutte polyno- mial. The first is an extension of the Tutte polynomial, from the class of matroids to that of polymatroids.