Class Of Matroids Research Articles

The Extremal Functions of Classes of Matroids of Bounded Branch-Width

For a set of matroids M, let ex M (n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field $$\mathbb{F}$$ , if M is a minor-closed class of $$\mathbb{F}$$ -representable matroids of bounded branch-width, then limn→ ∞ex M (n)/n exists and is a rational number, ∆. We also show that ex M (n) - ∆n is periodic when n is sufficiently large and that exM is achieved by a subclass of M of bounded path-width.

Theta rank, levelness, and matroid minors

The Theta rank of a finite point configuration V is the maximal degree necessary for a sum-of-squares representation of a non-negative affine function on V. This is an important invariant for polynomial optimization that is in general hard to determine. We study the Theta rank and levelness, a related discrete-geometric invariant, for matroid base configurations. It is shown that the class of matroids with bounded Theta rank or levelness is closed under taking minors. This allows for a characterization of matroids with bounded Theta rank or levelness in terms of forbidden minors. We give the complete (finite) list of excluded minors for Theta-1 matroids which generalizes the well-known series-parallel graphs. Moreover, the class of Theta-1 matroids can be characterized in terms of the degree of generation of the vanishing ideal and in terms of the psd rank for the associated matroid base polytope. We further give a finite list of excluded minors for k-level graphs and matroids and we investigate the graphs of Theta rank 2.

On the Combinatorics of Locally Repairable Codes via Matroid Theory

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters $(n,k,d,r,\delta )$ of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by Gopalan et al. for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters $(n,k,d,r,\delta )$ and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by Song et al.

The excluded minors for the class of matroids that are graphic or bicircular lift

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph G, the circuits of its bicircular lift matroid L(G) are the edge sets of those subgraphs of G that contain at least two cycles, and are minimal with respect to this property. For each cycle C of G, since L(G)/C is graphic and most graphic matroids are not bicircular lift, the class of bicircular lift matroids is not minor-closed. In this paper, we prove that the class of matroids that are graphic or bicircular lift has a finite list of excluded minors.

Representations of Bicircular Lift Matroids

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. The main result of this paper is a characterization of when two graphs give rise to the same bicircular lift matroid, which answers a question proposed by Irene Pivotto. In particular, aside from some appropriately defined "small" graphs, two graphs have the same bicircular lift matroid if and only if they are $2$-isomorphic in the sense of Whitney.

Toric ideals of series and parallel connections of matroids

In 1980, White conjectured that the toric ideal associated to a matroid is generated by binomials corresponding to a symmetric exchange. In this paper, we prove that classes of matroids for which the toric ideal is generated by quadrics and that has quadratic Gröbner bases, are closed under series and parallel extensions, series and parallel connections, and 2-sums.

Dendriform structures for restriction-deletion and restriction-contraction matroid Hopf algebras

We endow the set of isomorphism classes of matroids with a new Hopf algebra structure, in which the coproduct is implemented via the combinatorial operations of restriction and deletion. We also initiate the investigation of dendriform coalgebra structures on matroids and introduce a monomial invariant which satisfy a convolution identity with respect to restriction and deletion.

Stabilizer theorems for even cycle matroids

A signed graph is a representation of an even cycle matroid M if the cycles of M correspond to the even cycles of that signed graph. Two long standing open questions regarding even cycle matroids are the problem of finding an excluded minor characterization and the problem of efficiently recognizing this class of matroids. Progress on these problems has been hampered by the fact that even cycle matroids can have an arbitrary number of pairwise inequivalent representations (for a natural definition of equivalence). We show that we can bound the number of inequivalent representations of an even cycle matroid M (under some mild connectivity assumptions) if M contains any fixed size minor that is not a projection of a graphic matroid. For instance, any connected even cycle matroid which contains R10 as a minor has at most 6 inequivalent representations.

The Structure of $\{U_{2,5}, U_{3,5}\}$-Fragile Matroids

Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is strictly $\mathcal{N}$-fragile if $M$ has a member of $\mathcal{N}$ as minor and, for all $e \in E(M)$, at least one of $M\hspace{-0.5pt}\backslash e$ and $M/e$ has no minor in $\mathcal{N}$. In this paper we give a structural description of the strictly $\{U_{2,5},U_{3,5}\}$-fragile matroids that have six inequivalent representations over $\mathrm{GF}(5)$. Roughly speaking, these matroids fall into two classes. The matroids without an $\{X_8, Y_8, Y_8^{*}\}$-minor are constructed, up to duality, from one of two matroids by gluing wheels onto specified triangles. On the other hand, those matroids with an $\{X_8, Y_8, Y_8^{*}\}$-minor can be constructed from a matroid in $\{X_8, Y_8, Y_8^{*}\}$ by repeated application of elementary operations, and are shown to have path width 3. The characterization presented here will be crucial in finding the explicit list of excluded minors for two classes of matroids: the Hydra-5-representable matroids and the 2-regular matroids.

Positroids and non-crossing partitions

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on [ n ] [n] is connected equals 1 / e 2 1/e^2 asymptotically.

Projective geometries in exponentially dense matroids. I

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+1) uniform matroids, then there exists an integer n such that either every rank-r matroid in M can be covered by at most rn sets of rank at most a, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid in M can be covered by at most rnqr sets of rank at most a. This determines the maximum density of the matroids in M up to a polynomial factor.

Projective geometries in exponentially dense matroids. II

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+1) uniform matroids, then there exists an integer c such that either every rank-r matroid in M can be covered by at most rc rank-a sets, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid in M can be covered by at most cqr rank-a sets. In the latter case, this determines the maximum density of matroids in M up to a constant factor.

A Matroidal Framework for Network-Error Correcting Codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. A particularly interesting feature of this development is the ability to construct (scalar and vector) linearly solvable networks using certain classes of matroids. Furthermore, it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios. The current work attempts to establish a connection between matroid theory and network-error correcting and detecting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of linear network-error detecting codes to arrive at the definition of a matroidal error detecting network (and similarly, a matroidal error correcting network abstracting from network-error correcting codes). An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error detecting (correcting) network code if and only if it is a matroidal error detecting (correcting) network associated with a representable matroid. Therefore, constructing such network-error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa. We then present algorithms that enable the construction of matroidal error detecting and correcting networks with a specified capability of network-error correction. Using these construction algorithms, a large class of hitherto unknown scalar linearly solvable networks with multisource, multicast, and multiple-unicast network-error correcting codes is made available for theoretical use and practical implementation, with parameters, such as number of information symbols, number of sinks, number of coding nodes, error correcting capability, and so on, being arbitrary but for computing power (for the execution of the algorithms). The complexity of the construction of these networks is shown to be comparable with the complexity of existing algorithms that design multicast scalar linear network-error correcting codes. Finally, we also show that linear network coding is not sufficient for the general network-error correction (detection) problem with arbitrary demands. In particular, for the same number of network errors, we show a network for which there is a nonlinear network-error detecting code satisfying the demands at the sinks, whereas there are no linear network-error detecting codes that do the same.

The Highly Connected Matroids in Minor-Closed Classes

For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the structural characterization.

Unavoidable Minors of Large 4-Connected Bicircular Matroids

It is known that any 3-connected matroid that is large enough is certain to contain a minor of a given size belonging to one of a few special classes of matroids. This paper proves a similar unavoidable minor result for large 4-connected bicircular matroids. The main result follows from establishing the list of unavoidable minors of large 4-biconnected graphs, which are the graphs representing the 4-connected bicircular matroids. This paper also gives similar results for internally 4-connected and vertically 4-connected bicircular matroids.

Counting matroids in minor-closed classes

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on n elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an N-minor is asymptotically small in case N is one of the sparse paving matroids U2,k, U3,6, P6, Q6 or R6, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without an M(K4)-minor which asymptotically matches the best known lower bound on the number of all matroids, due to Knuth.

Generic and special constructions of pure O -sequences

It is shown that the h-vectors of Stanley-Reisner rings of three classes of matroids are pure O-sequences. The classes are (a) matroids that are truncations of other matroids, or more generally of Cohen-Macaulay complexes, (b) matroids whose dual is (rank + 2)-partite, and (c) matroids of Cohen-Macaulay type at most five. Consequences for the computational search for a counterexample to a conjecture of Stanley are discussed.

Matroid Secretary for Regular and Decomposable Matroids

In the matroid secretary problem we are given a stream of elements in random order and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream, then we can never get rid of it, and if we choose not to accept it, then we cannot later add it. Babaioff, Immorlica, and Kleinberg [Proceedings of SODA'07] introduced this problem, gave O(1)-competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)-competitive algorithm. However, most matroids that are known to admit an O(1)-competitive algorithm can be easily represented using graphs (e.g., graphic, cographic, and transversal matroids). In particular, there is very little known about $F$-representable matroids (the class of matroids that can be represented as elements of a vector space over a field $F$), which are one of the foundational types of matroids. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving O(1)-competitive algorithms for regular matroids (the class of matroids that are representable over any field) and use techniques that are fundamentally matroid-theoretic rather than graph-theoretic. Our main technique is to leverage the seminal regular matroid decomposition theorem of Seymour, which gives a method for decomposing any regular matroid into matroids which are graphic, cographic, or isomorphic to a simple 10-element matroid. We show how to combine in a black-box manner any algorithms for these basic classes into an algorithm for a given regular matroid, i.e., how to respect the decomposition. In fact, this allows us to generalize beyond regular matroids to any class of matroids that admits such a decomposition into classes for which we already have good algorithms. In particular, we give an O(1)-competitive algorithm for the class of max-flow min-cut matroids, which Seymour showed can be decomposed into regular matroids and copies of the Fano matroid.

Complete monotonicity for inverse powers of some combinatorially defined polynomials

We prove the complete monotonicity on (0,∞)n for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that P-β is completely monotone on a convex cone C: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of P-β for some β>0 can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for P, and is also related to the Rayleigh property for matroids.

Positroids, non-crossing partitions, and positively oriented matroids

We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals $1/e^2$ asymptotically. We also prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or <i>positive MacPhersonian</i>) is homeomorphic to a closed ball.