Rough set theory provides an effective tool to deal with uncertain, granular, and incomplete knowledge in information systems. Matroid theory generalizes the linear independence in vector spaces and has many applications in diverse fields, such as combinatorial optimization and rough sets. In this paper, we construct a matroidal structure of the generalized rough set based on a tolerance relati...

Oriented matroids have long been of use in various areas of combinatorics [BLS93]. Gelfand and MacPherson [GM92] initiated the use of oriented matroids in manifold and bundle theory, using them to formulate a combinatorial formula for the rational Pontrjagin classes of a differentiable manifold. MacPherson [Mac93] abstracted this into a manifold theory (combinatorial differential (CD) manifolds...

1. The Scope of These Talks 1 2. Matroid Theory Background 2 2.1. Basic Concepts 3 2.2. New Matroids from Old 13 2.3. Representations of Matroids over Fields 18 2.4. Projective and Affine Geometries 23 3. A First Taste of Extremal Matroid Theory: Cographic Matroids 25 4. Excluding Subgeometries: The Bose-Burton Theorem 28 5. Excluding the (q + 2)-Point Line as a Minor 43 6. Excluding F7 as a Mi...

This is a survey of algorithmic results in the theory of “discrete convex analysis” for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, the Fenchel min-max duality, and separation theorems. The technical development is based on matroid-theoretic concepts, in ...

Matroid theory is a generalization of the idea of linear independence. A matroid M consists of a finite set E (called the ground set) and a collection S of subsets of E satsifying the following conditions: (1) ∅ ∈ S; (2) if I ∈ S, then every subset of I is in S; (3) if I1 and I2 are in S and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ e is in S. The elements of S are calle...

Dominic Welsh began writing papers in matroid theory almost forty years ago. Since then, he has made numerous important contributions to the subject. His book Matroid Theory provided the first comprehensive treatment of the subject and has served as an invaluable reference to many workers in the field. Dominic’s work on matroids has been characterized by his ability to build bridges between the...

We observe that for planar graphs, the geometric duality relation generates both 2-isomorphism and abstract duality. This observation has surprising consequence links, equivalence defined by isomorphisms of checkerboard graphs is same as 2-isomorphisms graphs.

A matroid pencil is a pair of linking systems having the same ground sets in common. It provides a combinatorial abstraction of matrix pencils. This paper investigates the properties of matroid pencils analogous to the theory of Kronecker canonical form. As an application, we give a simple alternative proof for a theorem of Murota on power products of linking systems.

In this lecture, the focus is on submodular function in combinatorial optimizations. The first class of submodular functions which was studied thoroughly was the class of matroid rank functions. The flourishing stage of matroid theory came with Jack Edmonds’ work in 1960s, when he gave a minmax formula and an efficient algorithm to the matroid partition problem, from which the matroid intersect...