Matroid Research Articles

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.

Parameter-free methods distinguish Wnt pathway models and guide design of experiments

The canonical Wnt signaling pathway, mediated by β-catenin, is crucially involved in development, adult stem cell tissue maintenance, and a host of diseases including cancer. We analyze existing mathematical models of Wnt and compare them to a new Wnt signaling model that targets spatial localization; our aim is to distinguish between the models and distill biological insight from them. Using Bayesian methods we infer parameters for each model from mammalian Wnt signaling data and find that all models can fit this time course. We appeal to algebraic methods (concepts from chemical reaction network theory and matroid theory) to analyze the models without recourse to specific parameter values. These approaches provide insight into aspects of Wnt regulation: the new model, via control of shuttling and degradation parameters, permits multiple stable steady states corresponding to stem-like vs. committed cell states in the differentiation hierarchy. Our analysis also identifies groups of variables that should be measured to fully characterize and discriminate between competing models, and thus serves as a guide for performing minimal experiments for model comparison.

Note on the characterization and reduction of concept lattices through matroid theory

Mao[Characterization and reduction of concept lattices through matroid theory, Information Sciences 281, 10 (2014), 338–354] claims to make contributions to the study of concept lattices by using matroids. This note shows that his results are either trivial or wrong.

Completion and decomposition of a clutter into representable matroids

This paper deals with the question of completing a monotone increasing family of subsets Γ of a finite set Ω to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we prove that such vectorial completions of the family of subsets Γ exist and, in addition, we show that the minimal vectorial completions of the family Γ provide a decomposition of the clutter Λ of the inclusion-minimal elements of Γ. The computation of such vectorial decomposition of clutters is also discussed in some cases.

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.

A Bound for the Growth Rate of Binary Matroids Having No PG(k-1, 2)-Minor

Let M be a simple matroid having no U2, l + 2-minor. We show that if \({|E(M)| \geq {l}^{3k^3} {2}^(l+1)^{2^k} \left({\begin{array}{c} r(M)+1\\ 2 \end{array}}\right)}\) then M contains a simple minor N where \({{r(N) \geq k}}\) and \({{|E(N)| \geq 2^{r(N)} - 1}}\) . In particular, if M is a simple binary matroid where \({|E(M)| \geq {2}^{3^{2^k} +3k^{3}} \left( {\begin{array}{c} r(M)+1\\ 2 \end{array}}\right)}\) , then M contains a PG(k–1, 2)-minor.

Special Issue in Honour of James Oxley

This special issue is dedicated to Professor James Oxley to celebrate his sixtieth birthday in February 2013. Most of the papers are on topics in matroid theory, a subject to which James has contributed so much in his research career.

The cobases of the element splitting and the es-splitting matroids

In this paper, I characterize the cobases of the element splitting matroid M x,y ′ and the es-splitting matroid M x,y e in terms of the cobases of the original matroid M.

Matroids Denser than a Projective Geometry

The growth-rate function for a minor-closed class $\mathcal{M}$ of matroids is the function $h$ where, for each nonnegative integer $r$, $h(r)$ is the maximum number of elements of a simple matroid in $\mathcal{M}$ with rank at most $r$. The growth-rate theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential with some prime power $q$ as the base, or infinite. Moreover, if the growth-rate function is exponential with base $q$, then the class contains all GF$(q)$-representable matroids, and so $h(r)\ge \frac{q^r-1}{q-1}$ for each $r$. We characterise the classes that satisfy $h(r) = \frac{q^r-1}{q-1}$ for all sufficiently large $r$. As a consequence, we determine the eventual value of the growth-rate function for most classes defined by excluding lines, free spikes, or free swirls.

The algebraic combinatorial approach for low-rank matrix completion

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic ...

Dyck path triangulations and extendability (extended abstract)

We introduce the Dyck path triangulation of the cartesian product of two simplices $\Delta_{n-1}\times\Delta_{n-1}$. The maximal simplices of this triangulation are given by Dyck paths, and its construction naturally generalizes to produce triangulations of $\Delta_{r\ n-1}\times\Delta_{n-1}$ using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever$m\geq k>n$, any triangulations of $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ extends to a unique triangulation of $\Delta_{m-1}\times\Delta_{n-1}$. Moreover, with an explicit construction, we prove that the bound $k>n$ is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory. Nous introduisons la triangulation par chemins de Dyck du produit cartésien de deux simplexes $\Delta_{n-1}\times\Delta_{n-1}$. Les simplexes maximaux de cette triangulation sont donnés par des chemins de Dyck, et cette construction se généralise de façon naturelle pour produire des triangulations $\Delta_{r\ n-1}\times\Delta_{n-1}$ qui utilisent des chemins de Dyck rationnels. Notre étude de la triangulation par chemins de Dyck est motivée par des problèmes de prolongement de triangulations partielles de produits de deux simplexes. On montre que $m\geq k>n$ alors toute triangulation de $\Delta_{m-1}^{(k-1)}\times\Delta_{n-1}$ se prolonge en une unique triangulation de $\Delta_{m-1}\times\Delta_{n-1}$. De plus, avec une construction explicite, nous montrons que la borne $k>n$ est optimale. Nous présentons aussi des interprétations de nos résultats dans le langage des matroïdes orientés tropicaux, qui sont analogues aux résultats classiques de la théorie des matroïdes orientés.

On the Use of Matroid Theory for Distributed Cyber–Physical-Constrained Generator Scheduling in Smart Grid

Power systems are on the cusp of a rapid tech- nological, economic, and environmental evolution. Classi- cal problems considered by the power community must naturally adapt to new requirements. Physical and compu- tational constraints within a new landscape must be revis- ited. In this paper, we focus on the generator scheduling problem, which is also known as the unit commitment problem, in which we incorporate the new constraints of transmission line capacity limits and policy to give a more comprehensive view for planning in a smart grid. Moreover, we consider implications of our formulation to solution complexity. We introduce the concept of matroid theory to model unit commitment as a combinatorial problem and propose distributed solutions to facilitate optimal and correct generator scheduling. We account for communi- cations and computational complexity and demonstrate how although the constrained generator scheduling prob- lem is NP-hard, simpler versions of the problem lead to polynomial-time solutions.

Representability of matroids with a large projective geometry minor

We prove that, for each prime power q, there is an integer n such that if M is a 3-connected, representable matroid with a PG(n−1,q)-minor and no U2,q2+1-minor, then M is representable over GF(q). We also show that for ℓ≥2, if M is a 3-connected, representable matroid of sufficiently high rank with no U2,ℓ+2-minor and |E(M)|≥2ℓr(M)/2, then M is representable over a field of order at most ℓ.

The minimum number of minimal codewords in an [formula omitted]-code and in graphic codes

We survey some lower bounds on the function in the title based on matroid theory and address the following problem by Dosa et al. (2004): Determine the smallest number of circuits in a loopless matroid with no parallel elements and with a given size and rank. In the graphic 3-connected case we provide a lower bound which is a product of a linear function of the number of vertices and an exponential function of the average degree. We also prove that, for p≥38, every 3-connected graph with p vertices has at least as many cycles as the wheel with p vertices.

New Results on Ideal Multipartite Secret Sharing and its Applications to Group Communications

With the rapid development of various group-oriented services, multipartite group communications occur frequently in a single network, where a multipartite access structure is defined to be a collection of the subsets of users who may come from different parts of the network such that only users in an authorized subset of users can use their shares to build up a group key for a secure group communication. Most existing group key establishment schemes based on a secret sharing target on building up a group key for a threshold access structure, and need to compute a $$t$$t-degree interpolating polynomial in order to encrypt and decrypt the secret group key. This approach is not suitable and inefficient in terms of computational complexity for multipartite group environments which need to realize the multipartite access structures. In 1991, Brickell et al. proved that an ideal access structure is induced by a matroid and furthermore, an access structure is ideal if it is induced by a representable matroid. In this paper, we study the characterization of representable matroids. By using the connection between ideal secret sharing and matroids and, in particular, the recent results on ideal multipartite access structures and the connection between multipartite matroids and discrete polymatroids, we introduce a new concept on $$R$$R-tuple, which is determined by the rank function of the associated discrete polymatroid. Using this new concept, we come up a new and simple sufficient condition for a multipartite matroid to be representable (in fact, every matroid and every access structure are multipartite). In other words, we have developed a sufficient condition for an access structure to be ideal. These new results can be applied to establish multipartite group keys efficiently in secure group communications.

Closed-set lattice and modular matroid induced by covering-based rough sets

Covering is a common form of data representation, and covering-based rough sets, a technique of granular computing, provide an effective tool to deal with this type of data. However, many important problems of covering-based rough sets, such as covering reduction, are NP-hard so that most algorithms to solve them are greedy ones. Matroid theory, based on linear algebra and graph theory, provides well-established platforms for greedy algorithms. Lattice has been widely used in diverse fields, especially algorithm design, which plays an important role in covering reduction. Therefore, it is necessary to integrate covering-based rough sets with matroid and lattice. In this paper, we construct three types of matroids through covering-based rough sets and investigate their modularity. Moreover, we investigate some characteristics of these types of closed-set lattices induced by these three types of matroids and the relationships among these closed-set lattices. First, based on covering-based rough sets, three families of sets are constructed and proved to satisfy independent set axiom of matroids. So three types of matroids are induced by covering-based rough sets in this way. Second, some characteristics of these matroids, such as rank function, closure operator and closed set, are presented. Moreover, we investigate the characteristics of these closed-set lattices induced by these three types of matroids, such as modular pair, modular element. Finally, the relationships among these closed-set lattices induced by these three types of matroids are investigated. Especially, we prove that these three types of matroids induced by covering-based rough sets are all modular matroids.

Dyck path triangulations and extendability

We introduce the Dyck path triangulation of the cartesian product of two simplices Δn−1×Δn−1. The maximal simplices of this triangulation are given by Dyck paths, and the construction naturally generalizes to certain rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever m≥k>n, any triangulation of the product of the k-skeleton of Δm−1 with Δn−1 extends to a unique triangulation of Δm−1×Δn−1. Moreover, using the Dyck path triangulation, we prove that the bound k>n is optimal. We also exhibit interpretations of our results in the language of tropical oriented matroids that are analogous to classical results in oriented matroid theory.

Matroid intersection, base packing and base covering for infinite matroids

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and simplify the proofs of some cases where these conjectures were known to be true.

On the number of matroids

We consider the problem of determining mn, the number of matroids on n elements. The best known lower bound on mn is due to Knuth (1974) who showed that log log mn is at least n − 3/2 log n−O(1). On the other hand, Piff (1973) showed that log log mn ≤ n − log n + log log n + O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth's bound.We show that this is indeed the case, and prove an upper bound on log log mn that is within an additive 1 + o(1) term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.

Higher-Order CIS Codes

We introduce complementary information set codes of higher order. A binary linear code of length tk and dimension k is called a complementary information set code of order t (t-CIS code for short) if it has t pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a t-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 3 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length ≤ 12 is given. Nonlinear codes better than linear codes are derived by taking binary images of Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> -codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate 1/t, it either provides t disjoint information sets or proves that the code is not t-CIS. Using this algorithm, all optimal or best known [tk, k] codes, where t = 3, 4, . . . , 256 and 1≤ k ≤⌊256/t⌋ are shown to be t-CIS for all such k and t, except for t = 3 with k = 44 and t = 4 with k = 37.