A New Description of Transversal Matroids Through Rough Set Approach

Matroid theory is a useful tool for the combinatorial optimization issue in data mining, machine learning and knowledge discovery. Recently, combining matroid theory with rough sets is becoming interesting. In this paper, rough set approaches are used to investigate an important class of matroids, transversal matroids. We first extend the concept of upper approximation number functions in rough set theory and propose the notion of generalized upper approximation number functions on a set system. By means of the new notion, we give some necessary and sufficient conditions for a subset to be a partial transversal of a set system. Furthermore, we obtain a new description of a transversal matroid by the generalized upper approximation number function. We show that a transversal matroid can be induced by the generalized upper approximation number function which can be decomposed into the sum of some elementary generalized upper approximation number functions. Conversely, we also prove that a generalized upper approximation number function can induce a transversal matroid. Finally, we apply the generalized upper approximation number function to study the relationship among transversal matroids.

Lattices Related to Extensions of Presentations of Transversal Matroids

For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the ordered set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have presentations that extend $\mathcal{A}$; extensions are ordered by the weak order. We show that $T_{\mathcal{A}}$ is a distributive lattice, and that each finite distributive lattice is isomorphic to $T_{\mathcal{A}}$ for some presentation $\mathcal{A}$ of some transversal matroid $M$. We show that $T_{\mathcal{A}}\cap T_{\mathcal{B}}$, for any two presentations $\mathcal{A}$ and $\mathcal{B}$ of $M$, is a sublattice of both $T_{\mathcal{A}}$ and $T_{\mathcal{B}}$. We prove sharp upper bounds on $|T_{\mathcal{A}}|$ for presentations $\mathcal{A}$ of rank less than $r(M)$ in the order on presentations; we also give a sharp upper bound on $|T_{\mathcal{A}}\cap T_{\mathcal{B}}|$. The main tool we introduce to study $T_{\mathcal{A}}$ is the lattice $L_{\mathcal{A}}$ of closed sets of a certain closure operator on the lattice of subsets of $\{1,2,\ldots,r(M)\}$.

Infinite Gammoids: Minors and Duality

This sequel to Afzali Borujeni et. al. (2015) considers minors and duals of infinite gammoids. We prove that the class of gammoids defined by digraphs not containing a certain type of substructure, called an outgoing comb, is minor-closed. Also, we prove that finite-rank minors of gammoids are gammoids. Furthermore, the topological gammoids of Carmesin (2014) are proved to coincide, as matroids, with the finitary gammoids. A corollary is that topological gammoids are minor-closed.It is a well-known fact that the dual of any finite strict gammoid is a transversal matroid. The class of strict gammoids defined by digraphs not containing alternating combs, introduced in Afzali Borujeni et. al. (2015), contains examples which are not dual to any transversal matroid. However, we describe the duals of matroids in this class as a natural extension of transversal matroids. While finite gammoids are closed under duality, we construct a strict gammoid that is not dual to any gammoid.

Generalized Permutohedra, h-Vectors of Cotransversal Matroids and Pure O-Sequences

Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside an r-dimensional convex polytope and bases of a rank r transversal matroid.

Generalized permutohedra, h-vectors of cotransversal matroids and pure O-sequences (extended abstract)

International audience Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a $r$-dimensional convex polytope and bases of a rank $r$ transversal matroid. Stanley a conjecturé que le h-vecteur d'un complexe matroïde est une pure O-séquence. Nous allons le prouver pour les matroïdes cotransversaux en utilisant generalized permutohedra. Nous construisons une bijection entre les points du réseau intérieur d'un polytope convexe $r$-dimensions et les bases d'un matroïde transversal $r$-rang.

A Characterisation of Strict Matching Matroids

It is well known that a matroid is a transversal matroid if and only if it is a matching matroid (in the sense that it is the restriction of the matching structure of some graph to a subset of its vertices). A simple proof of that result is now known and in this paper it is used to answer the long-standing question of which transversal matroids are “strict” matching matroids; i.e. actually equal to the matching structure of a graph. We develop a straightforward test of “coloop-surfeit” that can be applied to any transversal matroid, and our main theorem shows that a transversal matroid is a strict matching matroid if and only if it has even rank and coloop-surfeit. Furthermore, the proofs are algorithmic and enable the construction of an appropriate graph from any presentation of a strict matching matroid.