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...

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the termin...

We show that the excluded minors for the class of matroids that are binary or ternary are U2,5, U3,5, U2,4⊕F7, U2,4⊕F ∗ 7 , U2,4⊕2F7, U2,4 ⊕2 F ∗ 7 , and the unique matroids obtained by relaxing a circuithyperplane in either AG(3, 2) or T12. The proof makes essential use of results obtained by Truemper on the structure of almost-regular matroids.

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given k weighted-matroids on same ground set. Our goal is to find a feasible partition that minimizes (maximizes) value of an function. A typical maximum over all subsets total weights elements in subset, which extensively studied scheduling literature. Likewise, as function, handle ...

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...

The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field k. If a matroid M is isomorphic to an algebraic matroid the latter is called an algebraic representation of M. Vector representations of matroids are defined similarly. A matroid may have algebraic (resp. vector) representations over fields of different characteristics. The problem in...

In this paper, we introduce the problem of Matroid-Constrained Vertex Cover: given a graph with weights on edges and matroid imposed vertices, our is to choose subset vertices that independent in matroid, objective maximizing total weight covered edges. This generalization much studied max $k$-vertex cover problem, which simple uniform it also special case monotone submodular function under con...