Counting Bases of Representable Matroids

Parameterized Counting of Trees, Forests and Matroid Bases

We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with k edges are #W[1]-hard when parameterized by k. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for k-forests implies #W[1]...

Beyond matroids? Permutation group bases and Boolean representable complexes

On Representable Matroids Having Neither U2,5– Nor U3,5–minors

Consider 3–connected matroids that are neither binary nor ternary and have neither U2,5– nor U3,5–minors: for example, AG(3, 2)′, the matroid obtained by relaxing a circuit-hyperplane of AG(3, 2). The main result of the paper shows that no matroid of this sort is representable over any field. This result makes it possible to extend known characterisations of the binary and ternary matroids repr...

On Representable Matroids and Ideal Secret Sharing

In secret sharing, the exact characterization of ideal access structures is a longstanding open problem. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have attracted a lot of attention. Due to the difficulty of finding general results, the characterizati...

BASES AND CIRCUITS OF FUZZIFYING MATROIDS

In this paper, as an application of fuzzy matroids, the fuzzifying greedy algorithm is proposed and an achievableexample is given. Basis axioms and circuit axioms of fuzzifying matroids, which are the semantic extension for thebasis axioms and circuit axioms of crisp matroids respectively, are presented. It is proved that a fuzzifying matroidis equivalent to a mapping which satisfies the basis ...