Rough matroids based on coverings

TL;DR

This paper introduces rough matroids based on coverings, exploring their properties and establishing connections between rough set theory and matroid theory, which could enhance algorithms in data analysis and optimization.

Contribution

It extends the concept of rough matroids to coverings, providing new theoretical insights and an equivalent formulation, building on previous work by Zhu and Wang.

Findings

The set of definable sets forms a lattice under inclusion.

Rough matroids based on coverings generalize those based on relations.

An equivalent formulation of rough matroids based on coverings is provided.

Abstract

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 $\subseteq$, 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.