Gaussian, Strong and Transversal Greedoids
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A Combinatorial Abstraction of the Shortest Path Problem and Its Relationship to Greedoids
A Combinatorial Abstraction of the Shortest Path Problem and its Relationship to Greedoids by E. Andrew Boyd Technical Report 88-7, May 1988 Abstract A natural generalization of the shortest path problem to arbitrary set systems is presented that captures a number of interesting problems, in cluding the usual graph-theoretic shortest path problem and the problem of finding a minimum weight set on a matroid. Necessary and sufficient conditions for the solution of this problem by the greedy algorithm are then investigated. In particular, it is noted that it is necessary but not sufficient for the underlying combinatorial structure to be a greedoid, and three ex tremely diverse collections of sufficient conditions taken from the greedoid literature are presented. 0.1 Introduction Two fundamental problems in the theory of combinatorial optimization are the shortest path problem and the problem of finding a minimum weight set on a matroid. It has long been recognized that both of these problems are solvable by a greedy algorithm - the shortest path problem by Dijk stra's algorithm [Dijkstra 1959] and the matroid problem by "the" greedy algorithm [Edmonds 1971]. Because these two problems are so fundamental and have such similar solution procedures it is natural to ask if they have a common generalization. The answer to this question not only provides insight into what structural properties make the greedy algorithm work but expands the class of combinatorial optimization problems known to be effi ciently solvable. The present work is related to the broader question of recognizing gen eral conditions under which a greedy algorithm can be used to solve a given combinatorial optimization problem. -
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. -
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). -
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. -
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. -
Factorisation of Greedoid Polynomials of Rooted Digraphs Arxiv
Factorisation of Greedoid Polynomials of Rooted Digraphs Kai Siong Yow∗1, Kerri Morgany2, and Graham Farrz1 1Faculty of Information Technology, Monash University, Clayton, Victoria 3800, Australia 2Deakin University, Geelong, Australia, School of Information Technology, Faculty of Science Engineering & Built Environment May 6, 2019 Abstract Gordon and McMahon defined a two-variable greedoid polynomial f(G; t; z) for any greedoid G. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polyno- mials of rooted digraphs have the multiplicative direct sum property. In ad- dition, these polynomials are divisible by 1 + z under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial f(D) of a rooted digraph D of order n GM-factorises if f(D) = f(G) · f(H) such that G and H are rooted digraphs of order at most n and f(G); f(H) =6 1. We study the GM-factorability of greedoid polynomials of rooted digraphs, particularly those that are not divisible by 1 + z. We give some examples and an infinite family of rooted digraphs that are not direct sums but their greedoid polynomials GM-factorise. Keywords: factorisation, greedoid polynomial, greedoid, directed branching gree- doid, rooted digraph, arborescence 1 Introduction arXiv:1809.02924v2 [math.CO] 3 May 2019 Greedoids were introduced by Korte and Lov´aszas collections of sets that generalise matroids [11]. Korte and Lov´aszobserved that the optimality of some \greedy" algo- rithms including breadth-first search could be traced back to an underlying combina- torial structure that satisfies the greedoid, but not the matroid, framework. -
A Greedoid Polynomial Which Distinguishes Rooted Arborescences
proceedings of the american mathematical society Volume 107, Number 2, October 1989 A GREEDOID POLYNOMIAL WHICH DISTINGUISHES ROOTED ARBORESCENCES GARY GORDON AND ELIZABETH MCMAHON (Communicated by Thomas H. Brylawski) Abstract. We define a two-variable polynomial fa(t, z) for a greedoid G which generalizes the standard one-variable greedoid polynomial A<j(f). Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of fG(t, z). We prove (Theorem 2.8) that when G is a rooted directed arborescence, fo(t, z) completely determines the arborescence. We also show the polynomial is irreducible over Z[t, z] for arborescences with only one edge directed out of the distinguished vertex. When G is a matroid, fc(t, z) coincides with the Tutte polynomial. We also give an example to show Theorem 2.8 fails for full greedoids. This example also shows fa(t, z) does not distinguish rooted arborescences among the class of all greedoids. 1. Introduction In this paper, we define a two-variable polynomial fG(t,z) for a greedoid G which generalizes the one-variable polynomial kG(t) given in [2, Section 9.6]. The main theorem states that when G is a rooted directed arborescence, fG{t,z) completely determines the arborescence. All the graphs in this paper are finite. In this section, we recall some definitions from graph theory and define the polynomial. A more complete account of the graph theory can be found in [7], for example. Definition 1.1. Let D = (V(D),E(D)) be a rooted directed graph (rooted di- graph), i.e., a directed graph with a distinguished vertex, denoted (*). -
The Complexity of the Matroid-Greedoid Partition Problem
The Complexity of the Matroid-Greedoid Partition Problem Vera Asodi¤ and Christopher Umansy Abstract We show that the maximum matroid-greedoid partition problem is NP-hard to approximate to within 1=2 + " for any " > 0, which matches the trivial factor 1/2 approximation algorithm. The main tool in our hardness of approximation result is an extractor code with polynomial rate, alphabet size and list size, together with an e±cient algorithm for list-decoding. We show that the recent extractor construction of Guruswami, Umans and Vadhan [5] can be used to obtain a code with these properties. We also show that the parameterized matroid-greedoid partition problem is ¯xed-parameter tractable. 1 Introduction Matroid theory is a general framework that captures a number of classical combinatorial optimization problems. Many natural problems can be formulated as matroid problems, including minimum weight spanning tree, maximum matching and various connectivity problems, and can be solved by general algorithms for matroids. For example, the problem of ¯nding a maximum matching in a bipartite graph is a special case of the maximum two matroid intersection problem. Edmonds [4] gave a polynomial time algorithm for this problem. A greedoid is a generalization of a matroid that captures even more optimization problems. Recall that a matroid is a set system M = (E; I), where the independent sets I ⊆ 2E have the following properties. (1) ; 2 I. (2) If X 2 I and Y ⊆ X then Y 2 I. (3) If X; Y 2 I and jXj > jY j then there is an element x 2 X n Y such that Y [ fxg 2 I. -
C Matroid Invariants and Greedoid Invariants 95
University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 2004 Greedoid invariants and the greedoid Tutte polynomial Chris Anne Clouse The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Clouse, Chris Anne, "Greedoid invariants and the greedoid Tutte polynomial" (2004). Graduate Student Theses, Dissertations, & Professional Papers. 9492. https://scholarworks.umt.edu/etd/9492 This Dissertation is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. Maureen and Mike MANSFIELD LIBRARY The University of Montana Permission is granted by the author to reproduce this material in its entirety, provided that this material is used for scholarly purposes and is properly cited in published works and reports. '*Please check "Yes" or "No" and provide signature Yes, I grant permission No, I do not grant permission Author's Signature: Date: 0 H Any copying for commercial purposes or financial gain may be undertaken only with the author's explicit consent. 8/98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Greedoid invariants and the greedoid Tutte polynomial by Chris Anne Clouse presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy The University of Montana May 2004 Approved by: ■person Dean, Graduate School Date Reproduced with permission of the copyright owner. -
TWO CHARACTERIZATIONS of ANTIMATROIDS* 1. Introduction
ANALELE S¸TIINT¸IFICE ALE UNIVERSITAT¸II˘ \AL.I. CUZA" DIN IAS¸I (S.N.) MATEMATICA,˘ Tomul LIX, 2013, f.2 DOI: 10.2478/v10157-012-0048-1 TWO CHARACTERIZATIONS OF ANTIMATROIDS* BY HUA MAO Abstract. NextClosure algorithm is a fast and good algorithm in formal concept analysis. With the assistance of NextClosure algorithm, this article provides a charac- terization of antimatroids. Additionally, this article introduces a characterization of k- truncated antimatroids. The characterization can be realized by an algorithm which works with polynomial time delay. Mathematics Subject Classification 2010: 05B35, 52A01, 68R05. Key words: antimatroid, k-truncated antimatroid, convex geometry, NextClosure algorithm, formal concept analysis. 1. Introduction Since 1940 (cf. [3]), when the antimatroiods were introduced, people have continuously searched for other characterizations of antimatroids (cf. [1], [2], [6], [8], [9], [11], [12]). According to our knowledge, we add up the following points. (α) Kempner and Levit point out in [8] that there are many equivalent axiomatizations of antimatroids that may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. They introduce in [8] an algorithmic characterization of antimatroids based on the idea of optimization using set functions defined as minimum values of linkages between a set and the elements from the set complement. *This research is supported by NSF of China (11101115, 61202178, 61073121) and NSF of Hebei Province (F2012402037, A2013201119). 454 HUA MAO 2 (β) There is an algorithmic characterization of antimatroids based on the language definition in algorithmic idea in [2]. (γ) There is one-to-one correspondence between antimatroids and con- vex geometries (cf. -
A Note on Ultrametric Spaces, Minimum Spanning Trees and the Topological Distance Algorithm
information Article A Note on Ultrametric Spaces, Minimum Spanning Trees and the Topological Distance Algorithm Jörg Schäfer Department of Computer Science and Engineering, Frankfurt University of Applied Sciences, Nibelungenplatz 1, D-60318 Frankfurt am Main, Germany; [email protected] Received: 9 July 2020; Accepted: 26 August 2020; Published: 28 August 2020 Abstract: We relate the definition of an ultrametric space to the topological distance algorithm—an algorithm defined in the context of peer-to-peer network applications. Although (greedy) algorithms for constructing minimum spanning trees such as Prim’s or Kruskal’s algorithm have been known for a long time, they require the complete graph to be specified and the weights of all edges to be known upfront in order to construct a minimum spanning tree. However, if the weights of the underlying graph stem from an ultrametric, the minimum spanning tree can be constructed incrementally and it is not necessary to know the full graph in advance. This is possible, because the join algorithm responsible for joining new nodes on behalf of the topological distance algorithm is independent of the order in which the nodes are added due to the property of an ultrametric. Apart from the mathematical elegance which some readers might find interesting in itself, this provides not only proofs (and clearer ones in the opinion of the author) for optimality theorems (i.e., proof of the minimum spanning tree construction) but a simple proof for the optimality of the reconstruction algorithm omitted in previous publications too. Furthermore, we define a new algorithm by extending the join algorithm to minimize the topological distance and (network) latency together and provide a correctness proof. -
The Bhargava Greedoid As a Gaussian Elimination Greedoid
The Bhargava greedoid as a Gaussian elimination greedoid Darij Grinberg* June 11, 2021 Abstract. Inspired by Manjul Bhargava’s theory of generalized facto- rials, Fedor Petrov and the author have defined the Bhargava greedoid –a greedoid (a matroid-like set system on a finite set) assigned to any “ul- tra triple” (a somewhat extended variant of a finite ultrametric space). Here we show that the Bhargava greedoid of a finite ultra triple is always a Gaussian elimination greedoid over any sufficiently large (e.g., infinite) field; this is a greedoid analogue of a representable matroid. We find necessary and sufficient conditions on the size of the field to ensure this. Contents 1. Gaussian elimination greedoids 4 1.1. Thedefinition ................................ 4 1.2. Context.................................... 5 2. V-ultra triples 6 3. The main theorem 9 4. Cliques and stronger bounds 10 arXiv:2001.05535v3 [math.CO] 10 Jun 2021 5. The converse direction 11 6. Valadic V-ultra triples 12 7. Isomorphism 25 8. Decomposing a V-ultra triple 26 *Drexel University, Korman Center, Room 291, 15 S 33rd Street, Philadelphia PA, 19104, USA 1 TheBhargavagreedoidasaGaussianeliminationgreedoid page 2 9. Valadic representation of V-ultra triples 30 10.Proof of the main theorem 36 11.Proof of Theorem 5.1 36 11.1.Closedballs ................................. 36 11.2. Exchange results for sets intersecting a ball . ........ 39 11.3. Gaussian elimination greedoids in terms of determinants ...... 42 11.4.Provingthetheorem ............................ 48 12.Appendix: Gaussianeliminationgreedoidsarestrong 51 13.Appendix: Proof of Proposition 1.7 55 14.Appendix: Proofs of some properties of L 58 *** The notion of a greedoid was coined in 1981 by Korte and Lóvász, and has since seen significant developments ([KoLoSc91], [BjoZie92]).