Graph and matrix approaches to rough sets through matroids

Rough sets are efficient for attribute reduction and rule extraction in data mining. However, many important problems including attribute reduction in rough sets are NP-hard, therefore the algorithms to solve them are often greedy. Matroids, generalized from linear independence in vector spaces, provide well-established platforms for greedy algorithm design. In this paper, we use graph and matrix approaches to study rough sets through matroids. First, we construct an isomorphism from equivalence relations to 2-circuit matroids, and then propose graph representations of lower and upper approximations through the graphic matroid. We also study graph representations of lower and upper approximations by that of the dual of the matroid. Second, in light of the fact that the relational matrix is a representable matrix of the matroid induced by an equivalence relation, matrix representations of lower and upper approximations are obtained with the representable matrix of the matroid. In a word, borrowing from matroids, this work presents two interesting views, graph and matrix ones, to investigate rough sets.