scholarly journals Equivalent Characterizations of Some Graph Problems by Covering-Based Rough Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shiping Wang ◽  
Qingxin Zhu ◽  
William Zhu ◽  
Fan Min
Keyword(s):  
Graph Theory ◽  
Rough Sets ◽  
Systematic Approach ◽  
Rough Set Theory ◽  
Approximation Number ◽  
Edge Cover ◽  
Upper Approximation ◽  
Graph Problems ◽  
New Concepts ◽  
Equivalent Characterization

Covering is a widely used form of data structures. Covering-based rough set theory provides a systematic approach to this data. In this paper, graphs are connected with covering-based rough sets. Specifically, we convert some important concepts in graph theory including vertex covers, independent sets, edge covers, and matchings to ones in covering-based rough sets. At the same time, corresponding problems in graphs are also transformed into ones in covering-based rough sets. For example, finding a minimal edge cover of a graph is translated into finding a minimal general reduct of a covering. The main contributions of this paper are threefold. First, any graph is converted to a covering. Two graphs induce the same covering if and only if they are isomorphic. Second, some new concepts are defined in covering-based rough sets to correspond with ones in graph theory. The upper approximation number is essential to describe these concepts. Finally, from a new viewpoint of covering-based rough sets, the general reduct is defined, and its equivalent characterization for the edge cover is presented. These results show the potential for the connection between covering-based rough sets and graphs.

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals Covering-Based Rough Sets on Eulerian Matroids

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bin Yang ◽  
Ziqiong Lin ◽  
William Zhu
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Closure Operator ◽  
Matroid Theory ◽  
Upper Approximation ◽  
Significant Generalization ◽  
Eulerian Matroid ◽  
Vital Structure

Rough set theory is an efficient and essential tool for dealing with vagueness and granularity in information systems. Covering-based rough set theory is proposed as a significant generalization of classical rough sets. Matroid theory is a vital structure with high applicability and borrows extensively from linear algebra and graph theory. In this paper, one type of covering-based approximations is studied from the viewpoint of Eulerian matroids. First, we explore the circuits of an Eulerian matroid from the perspective of coverings. Second, this type of covering-based approximations is represented by the circuits of Eulerian matroids. Moreover, the conditions under which the covering-based upper approximation operator is the closure operator of a matroid are presented. Finally, a matroidal structure of covering-based rough sets is constructed. These results show many potential connections between covering-based rough sets and matroids.

Model of Covering Rough Sets in the Machine Intelligence

2012 ◽  
Vol 548 ◽  
pp. 735-739
Author(s):  
Hong Mei Nie ◽  
Jia Qing Zhou
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Machine Intelligence ◽  
Upper Approximation ◽  
Generalized Rough Sets

Rough set theory has been proposed by Pawlak as a useful tool for dealing with the vagueness and granularity in information systems. Classical rough set theory is based on equivalence relation. The covering rough sets are an improvement of Pawlak rough set to deal with complex practical problems which the latter one can not handle. This paper studies covering-based generalized rough sets. In this setting, we investigate common properties of classical lower and upper approximation operations hold for the covering-based lower and upper approximation operations and relationships among some type of covering rough sets.

ON ATTRIBUTE REDUCTION WITH INTUITIONISTIC FUZZY ROUGH SETS

2012 ◽  
Vol 20 (01) ◽  
pp. 59-76 ◽  
Author(s):  
ZHIMING ZHANG ◽  
JINGFENG TIAN
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Theoretical Foundation ◽  
Attribute Reduction ◽  
Upper Approximation ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Rough Sets ◽  
Axiomatic Approaches

Intuitionistic fuzzy (IF) rough sets are the generalization of traditional rough sets obtained by combining the IF set theory and the rough set theory. The existing research on IF rough sets mainly concentrates on the establishment of lower and upper approximation operators using constructive and axiomatic approaches. Less effort has been put on the attribute reduction of databases based on IF rough sets. This paper systematically studies attribute reduction based on IF rough sets. Firstly, attribute reduction with traditional rough sets and some concepts of IF rough sets are reviewed. Then, we introduce some concepts and theorems of attribute reduction with IF rough sets, and completely investigate the structure of attribute reduction. Employing the discernibility matrix approach, an algorithm to find all attribute reductions is also presented. Finally, an example is proposed to illustrate our idea and method. Altogether, these findings lay a solid theoretical foundation for attribute reduction based on IF rough sets.

scholarly journals A Novel Method of the Generalized Interval-Valued Fuzzy Rough Approximation Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Tianyu Xue ◽  
Zhan’ao Xue ◽  
Huiru Cheng ◽  
Jie Liu ◽  
Tailong Zhu
Keyword(s):  
Rough Sets ◽  
Rough Set Theory ◽  
Binary Relations ◽  
Approximation Space ◽  
Upper Approximation ◽  
Approximation Operators ◽  
Rough Approximation ◽  
Fuzzy Binary Relation ◽  
Novel Method ◽  
Interval Valued

Rough set theory is a suitable tool for dealing with the imprecision, uncertainty, incompleteness, and vagueness of knowledge. In this paper, new lower and upper approximation operators for generalized fuzzy rough sets are constructed, and their definitions are expanded to the interval-valued environment. Furthermore, the properties of this type of rough sets are analyzed. These operators are shown to be equivalent to the generalized interval fuzzy rough approximation operators introduced by Dubois, which are determined by any interval-valued fuzzy binary relation expressed in a generalized approximation space. Main properties of these operators are discussed under different interval-valued fuzzy binary relations, and the illustrative examples are given to demonstrate the main features of the proposed operators.

Quantitative analysis for covering-based rough sets through the upper approximation number

2013 ◽  
Vol 220 ◽  
pp. 483-491 ◽  
Author(s):  
Shiping Wang ◽  
Qingxin Zhu ◽  
William Zhu ◽  
Fan Min
Keyword(s):  
Quantitative Analysis ◽  
Rough Sets ◽  
Approximation Number ◽  
Upper Approximation

A New Description of Transversal Matroids Through Rough Set Approach

2021 ◽  
Vol 179 (4) ◽  
pp. 399-416
Author(s):  
Zhaohao Wang
Keyword(s):  
Rough Set ◽  
Rough Set Theory ◽  
Sufficient Conditions ◽  
Mining Machine ◽  
Matroid Theory ◽  
Approximation Number ◽  
Upper Approximation ◽  
Transversal Matroid ◽  
Necessary And Sufficient ◽  
The Relationship

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.

scholarly journals Matroidal Structure of Generalized Rough Sets Based on Tolerance Relations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Hui Li ◽  
Yanfang Liu ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Matroid Theory ◽  
Upper Approximation ◽  
Tolerance Relation ◽  
The Family ◽  
Generalized Rough Set ◽  
The Relationship ◽  
Generalized Rough Sets

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 relation. First, a family of sets are constructed through the lower approximation of a tolerance relation and they are proved to satisfy the circuit axioms of matroids. Thus we establish a matroid with the family of sets as its circuits. Second, we study the properties of the matroid including the base and the rank function. Moreover, we investigate the relationship between the upper approximation operator based on a tolerance relation and the closure operator of the matroid induced by the tolerance relation. Finally, from a tolerance relation, we can get a matroid of the generalized rough set based on the tolerance relation. The matroid can also induce a new relation. We investigate the connection between the original tolerance relation and the induced relation.

scholarly journals Geometric Lattice Structure of Covering-Based Rough Sets through Matroids

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Aiping Huang ◽  
William Zhu
Keyword(s):  
Rough Sets ◽  
Rough Set Theory ◽  
Lattice Structure ◽  
Search Algorithm ◽  
Algorithm Design ◽  
Geometric Lattice ◽  
Lattice Structures ◽  
Upper Approximation ◽  
The Relationship ◽  
Geometric Lattices

Covering-based rough set theory is a useful tool to deal with inexact, uncertain, or vague knowledge in information systems. Geometric lattice has been widely used in diverse fields, especially search algorithm design, which plays an important role in covering reductions. In this paper, we construct three geometric lattice structures of covering-based rough sets through matroids and study the relationship among them. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by a covering. Then its characteristics, such as atoms, modular elements, and modular pairs, are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, we present three sufficient and necessary conditions for two types of covering upper approximation operators to be closure operators of matroids. We also represent two types of matroids through closure axioms and then obtain two geometric lattice structures of covering-based rough sets. Third, we study the relationship among these three geometric lattice structures. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely, geometric lattice, to study covering-based rough sets.

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