Matroids In Terms Research Articles

Sweeps, Polytopes, Oriented Matroids, and Allowable Graphs of Permutations

A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Sweeps of a point configuration are in bijection with faces of an associated sweep polytope. Mimicking the fact that sweep polytopes are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. Pseudo-sweeps are a generalization of sweeps in which the sweeping hyperplane is allowed to slightly change direction, and that can be extended to arbitrary oriented matroids in terms of cellular strings. We prove that for sweepable oriented matroids, sweep oriented matroids provide a sphere that is a deformation retract of the poset of pseudo-sweeps. This generalizes a property of sweep polytopes (which can be interpreted as monotone path polytopes of zonotopes), and solves a special case of the strong Generalized Baues Problem for cellular strings. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids.

Matroids over skew tracts

Matroids over tracts provide an algebraic framework simultaneously generalizing the notions of matroids, oriented matroids, and valuated matroids, presented by Baker and Bowler. Pendavingh partially extended this theory to skew hyperfields and presented a new axiom system in terms of quasi-Plücker coordinates. We present a theory of matroids over skew tracts, which generalizes both the theory of matroids over tracts and the theory of weak matroids over skew hyperfields developed by Pendavingh. We give several cryptomorphic axiom systems for such matroids in terms of circuits, quasi-Plücker coordinates and dual pairs.

Hypergraph characterization of split matroids

We provide a combinatorial study of split matroids, a class that was motivated by the study of matroid polytopes from a tropical geometry point of view. A nice feature of split matroids is that they generalize paving matroids, while being closed under duality and taking minors. Furthermore, these matroids proved to be useful in giving exact asymptotic bounds for the dimension of the Dressian, and also implied new results on the rays of the tropical Grassmannians.In the present paper, we introduce the notion of elementary split matroids, a subclass of split matroids that contains all connected split matroids. We give a hypergraph characterization of elementary split matroids in terms of independent sets, and show that the proposed class is closed not only under duality and taking minors but also truncation. We further show that, in contrast to split matroids, the proposed class can be characterized by a single forbidden minor. As an application, we provide a complete list of binary split matroids.

Unbreakable matroids

A matroid M is unbreakable if M is connected and, for each flat F, the matroid M/F is connected. Equivalently, M is unbreakable if its dual has no two skew circuits. This paper characterizes unbreakable matroids in terms of excluded parallel minors and determines all regular unbreakable matroids.

Even circuits in oriented matroids

In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we show that the problem of detecting an even directed circuit in a regular matroid is polynomially equivalent to the recognition of non-even oriented matroids. Our main result is a precise characterisation of the class of non-even oriented cographic matroids in terms of forbidden minors, which complements an existing characterisation of non-even oriented graphic matroids by Seymour and Thomassen.Mathematics Subject Classifications: 05B35, 05C20, 05C70, 05C75, 05C83, 05C85, 52C40

Rainbow and monochromatic circuits and cocircuits in binary matroids

Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if an n-element rank r binary matroid M is colored with exactly r colors, then M either contains a rainbow colored circuit or a monochromatic cocircuit. As the class of binary matroids is closed under taking duals, this immediately implies that if M is colored with exactly n−r colors, then M either contains a rainbow colored cocircuit or a monochromatic circuit. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids.Motivated by a conjecture of Bérczi, Schwarcz and Yamaguchi, we also analyze the relation between the covering number of a binary matroid and the maximum number of colors or the maximum size of a color class in any of its rainbow circuit-free colorings. For simple graphic matroids, we show that there exists a rainbow circuit-free coloring that uses each color at most twice only if the graph is (2,3)-sparse, that is, it is independent in the 2-dimensional rigidity matroid. Furthermore, we give a complete characterization of minimally rigid graphs admitting such a coloring.

Chow rings of vector space matroids

The Chow ring of a matroid (or more generally, atomic latice) is an invariant whose importance was demonstrated by Adiprasito, Huh and Katz, who used it to resolve the long-standing Heron-Rota-Welsh conjecture. Here, we make a detailed study of the Chow rings of uniform matroids and of matroids of finite vector spaces. In particular, we express the Hilbert series of such matroids in terms of permutation statistics; in the full rank case, our formula yields the maj-exc $q$-Eulerian polynomials of Shareshian and Wachs. We also provide a formula for the Charney-Davis quantities of such matroids, which can be expressed in terms of either determinants or $q$-secant numbers.

Matroidal Structure Based On Soft-Sets

A matroidal structure that generalizes the properties of independence. Relevant applications are found in graph theory and linear algebra. This paper will focus on the definitions of a matroid in terms of generalization for a crisp set called soft-set and soft-point, also we give some results related to this concept. A soft-matroid is defined and examples of soft-systems which form are given. The novel concept of independent soft-set is introduced. The notion maximal of independent soft-sets and minimal dependent soft-sets, with examples from linear algebra and soft-graph theory, are illustrated. Finally, we investigate some fundamental properties of soft-matroid.

The lattice and matroid representations of definable sets in generalized rough sets based on relations

The definable set is a core concept in rough set theory. It plays an important role in the characterizations of rough sets. In this paper, we study the lattice and matroid representations of definable sets in generalized rough sets based on relations. First, we propose the lower definable lattice, consisting of all lower definable sets with set inclusion order. Then we give some conditions under which the lower definable lattice is distributive (or geometric, or Boolean). Furthermore, we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations. On the one hand, we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive. On the other hand, we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation. Finally, we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice. We show that if a lower definable lattice is a lattice of closed sets of a matroid, then it must be an open-closed set lattice of a matroid. In addition, we prove that some lower definable lattices are not the lattices of closed sets of matroids. These results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations.

Matroids over partial hyperstructures

We present an algebraic framework which simultaneously generalizes the notion of linear subspaces, matroids, valuated matroids, oriented matroids, and regular matroids. To do this, we first introduce algebraic objects which we call tracts; they generalize both hyperfields in the sense of Krasner and partial fields in the sense of Semple and Whittle. We then define matroids over tracts; in fact, there are (at least) two natural notions of matroid in this general context, which we call weak and strong matroids. We give “cryptomorphic” axiom systems for such matroids in terms of circuits, Grassmann–Plücker functions, and dual pairs, and establish some basic duality results. We then explore sufficient criteria for the notions of weak and strong matroids to coincide. This is the case whenever vectors and covectors are orthogonal, and is closely related to the notion of “perfect fuzzy rings” from [15]. For example, if F is a particularly nice kind of tract called a doubly distributive partial hyperfield, we show that the notions of weak and strong matroid over F coincide. Our theory of matroids over tracts is closely related to but more general than “matroids over fuzzy rings” in the sense of Dress and Dress–Wenzel [12–15].

Stirling numbers in braid matroid Kazhdan–Lusztig polynomials

Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan–Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan–Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling numbers of the first kind and Stirling numbers of the second kind, as well as a non-recursive formula for the braid matroid Kazhdan–Lusztig polynomials.

Generalized matroids based on three-way decision models

Three-way decision theory is an extension of the commonly used binary-decision model with an added third option. It is originally introduced to explain the three regions of probabilistic rough sets. Every object in a three-way decision model can be assigned to one of the three regions according to its evaluation value under an evaluation function. This paper first introduces three-way decision models based on subset-evaluation which generalize the original models. By the axiomatic approach, we characterize a matroid in terms of evaluation function and then define three-way matroids based on this characterization. Furthermore, three-way matroids are generalized to three-way fuzzy matroids and an equivalent description of three-way fuzzy matroid in terms of fuzzy independent set system is presented. Finally, we give the second description of three-way fuzzy matroid: a three-way fuzzy matroid is exactly the greatest element of an equivalence class. Additionally, relations of notions introduced in this paper are also pointed out.

The Category of Matroids

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits.

A Generalized Polymatroid Approach to Stable Matchings with Lower Quotas

Classified stable matching, proposed by Huang, describes a matching model between academic institutes and applicants, in which each institute has upper and lower quotas on classes, i.e., subsets of applicants. Huang showed that the problem to decide whether there exists a stable matching or not is NP-hard in general. On the other hand, he showed that the problem is solvable if classes form a laminar family. For this case, Fleiner and Kamiyama gave a concise interpretation in terms of matroids and showed the lattice structure of stable matchings. In this paper we introduce stable matchings on generalized matroids, extending the model of Fleiner and Kamiyama. We design a polynomial-time algorithm which finds a stable matching or reports the nonexistence. We also show that the set of stable matchings, if nonempty, forms a lattice with several significant properties. Furthermore, we extend this structural result to the polyhedral framework, which we call stable allocations on generalized polymatroids.

Wilson Loop Diagrams and Positroids

In this paper, we study a new application of the positive Grassmannian to Wilson loop diagrams (or MHV diagrams) for scattering amplitudes in N= 4 Super Yang–Mill theory (N = 4 SYM). There has been much interest in studying this theory via the positive Grassmannians using BCFW recursion. This is the first attempt to study MHV diagrams for planar Wilson loop calculations (or planar amplitudes) in terms of positive Grassmannians. We codify Wilson loop diagrams completely in terms of matroids. This allows us to apply the combinatorial tools in matroid theory used to identify positroids (non-negative Grassmannians) to Wilson loop diagrams. In doing so, we find that certain non-planar Wilson loop diagrams define positive Grassmannians. While non-planar diagrams do not have physical meaning, this finding suggests that they may have value as an algebraic tool, and deserve further investigation.

Clones in 3-connected frame matroids

We determine the structure of clonal classes of 3-connected frame matroids in terms of the structure of biased graphs. Robbins has conjectured that a 3-connected non-uniform matroid with a clonal class of size q−1 is not GF(q)-representable. We confirm the conjecture for the class of frame matroids.

Finitary and cofinitary gammoids

A gammoid is a matroid defined using linkability of vertex sets in a (possibly infinite) digraph. Related types of matroids are strict gammoids and transversal matroids, three aspects of which will be considered as follows.First, we investigate the interaction between matroid properties of strict gammoids and transversal matroids and graphic properties of the defining graphs. In particular, we characterize cofinitary strict gammoids and cofinitary transversal matroids among, respectively, strict gammoids and transversal matroids in terms of the defining graphs.The set of finite circuits of a matroid defines the finitarization matroid on the same ground set. A matroid is nearly finitary if every base can be extended to a base of the finitarization matroid by adding finitely many elements. Aigner-Horev et al. (2011) [6] raised the question whether the number of such additions is bounded for a fixed nearly finitary matroid. We answer this question positively in the classes of strict gammoids and transversal matroids.Piff and Welsh (1970) proved the classical result that a finite strict gammoid/transversal matroid is representable over any large enough field. In this direction, we prove that finitary strict gammoids and transversal matroids are representable over some field, and hence the cofinitary counterparts are thin sums representable, a new notion of representability introduced by Bruhn and Diestel (2011).

Fixing Numbers for Matroids

Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism group). We prove the fixing numbers for the cycle matroid and bicircular matroid associated with 3-connected graphs are identical. Many of these results have interpretations through permutation groups, and we make this connection explicit.

Obstacles to Decomposition Theorems for Sixth-Root-of-Unity Matroids

We construct an infinite family of highly connected sixth-root-of-unity matroids that are not near-regular. This family is an obstacle to any decomposition theorem for sixth-rootof- unity matroids in terms of near-regular matroids.

A graph-theoretical axiomatization of oriented matroids

We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.