55 research outputs found
Matroidal structure of generalized rough sets based on symmetric and transitive relations
Rough sets are efficient for data pre-process in data mining. Lower and upper
approximations are two core concepts of rough sets. This paper studies
generalized rough sets based on symmetric and transitive relations from the
operator-oriented view by matroidal approaches. We firstly construct a
matroidal structure of generalized rough sets based on symmetric and transitive
relations, and provide an approach to study the matroid induced by a symmetric
and transitive relation. Secondly, this paper establishes a close relationship
between matroids and generalized rough sets. Approximation quality and
roughness of generalized rough sets can be computed by the circuit of matroid
theory. At last, a symmetric and transitive relation can be constructed by a
matroid with some special properties.Comment: 5 page
Geometric lattice structure of covering and its application to attribute reduction through matroids
The reduction of covering decision systems is an important problem in data
mining, and covering-based rough sets serve as an efficient technique to
process the problem. Geometric lattices have been widely used in many fields,
especially greedy algorithm design which plays an important role in the
reduction problems. Therefore, it is meaningful to combine coverings with
geometric lattices to solve the optimization problems. In this paper, we obtain
geometric lattices from coverings through matroids and then apply them to the
issue of attribute reduction. First, a geometric lattice structure of a
covering is constructed through transversal matroids. Then its atoms are
studied and used to describe the lattice. Second, considering that all the
closed sets of a finite matroid form a geometric lattice, we propose a
dependence space through matroids and study the attribute reduction issues of
the space, which realizes the application of geometric lattices to attribute
reduction. Furthermore, a special type of information system is taken as an
example to illustrate the application. In a word, this work points out an
interesting view, namely, geometric lattice to study the attribute reduction
issues of information systems
Some characteristics of matroids through rough sets
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 terminology of linear algebra and graph theory. We can combine rough set
theory with matroid theory through using rough sets to study some
characteristics of matroids. In this paper, we apply rough sets to matroids
through defining a family of sets which are constructed from the upper
approximation operator with respect to an equivalence relation. First, we prove
the family of sets satisfies the support set axioms of matroids, and then we
obtain a matroid. We say the matroids induced by the equivalence relation and a
type of matroid, namely support matroid, is induced. Second, through rough
sets, some characteristics of matroids such as independent sets, support sets,
bases, hyperplanes and closed sets are investigated.Comment: 13 page
Covering matroid
In this paper, we propose a new type of matroids, namely covering matroids,
and investigate the connections with the second type of covering-based rough
sets and some existing special matroids. Firstly, as an extension of
partitions, coverings are more natural combinatorial objects and can sometimes
be more efficient to deal with problems in the real world. Through extending
partitions to coverings, we propose a new type of matroids called covering
matroids and prove them to be an extension of partition matroids. Secondly,
since some researchers have successfully applied partition matroids to
classical rough sets, we study the relationships between covering matroids and
covering-based rough sets which are an extension of classical rough sets.
Thirdly, in matroid theory, there are many special matroids, such as
transversal matroids, partition matroids, 2-circuit matroid and
partition-circuit matroids. The relationships among several special matroids
and covering matroids are studied.Comment: 15 page
Characteristic of partition-circuit matroid through approximation number
Rough set theory is a useful tool to deal with uncertain, granular and
incomplete knowledge in information systems. And it is based on equivalence
relations or partitions. Matroid theory is a structure that generalizes linear
independence in vector spaces, and has a variety of applications in many
fields. In this paper, we propose a new type of matroids, namely,
partition-circuit matroids, which are induced by partitions. Firstly, a
partition satisfies circuit axioms in matroid theory, then it can induce a
matroid which is called a partition-circuit matroid. A partition and an
equivalence relation on the same universe are one-to-one corresponding, then
some characteristics of partition-circuit matroids are studied through rough
sets. Secondly, similar to the upper approximation number which is proposed by
Wang and Zhu, we define the lower approximation number. Some characteristics of
partition-circuit matroids and the dual matroids of them are investigated
through the lower approximation number and the upper approximation number.Comment: 12 page
Rough matroids based on coverings
The introduction of covering-based rough sets has made a substantial
contribution to the classical rough sets. However, many vital problems in rough
sets, including attribution reduction, are NP-hard and therefore the algorithms
for solving them are usually greedy. Matroid, as a generalization of linear
independence in vector spaces, it has a variety of applications in many fields
such as algorithm design and combinatorial optimization. An excellent
introduction to the topic of rough matroids is due to Zhu and Wang. On the
basis of their work, we study the rough matroids based on coverings in this
paper. First, we investigate some properties of the definable sets with respect
to a covering. Specifically, it is interesting that the set of all definable
sets with respect to a covering, equipped with the binary relation of inclusion
, constructs a lattice. Second, we propose the rough matroids based
on coverings, which are a generalization of the rough matroids based on
relations. Finally, some properties of rough matroids based on coverings are
explored. Moreover, an equivalent formulation of rough matroids based on
coverings is presented. These interesting and important results exhibit many
potential connections between rough sets and matroids.Comment: 15page
Parametric matroid of rough set
Rough set is mainly concerned with the approximations of objects through an
equivalence relation on a universe. Matroid is a combinatorial generalization
of linear independence in vector spaces. In this paper, we define a parametric
set family, with any subset of a universe as its parameter, to connect rough
sets and matroids. On the one hand, for a universe and an equivalence relation
on the universe, a parametric set family is defined through the lower
approximation operator. This parametric set family is proved to satisfy the
independent set axiom of matroids, therefore it can generate a matroid, called
a parametric matroid of the rough set. 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, since partition-circuit matroids were well
studied through the lower approximation number, 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.Comment: 15 page
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