Related papers: On Cyclic Matroids and their Applications
Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are…
There is a long list of open questions rooted in the same underlying problem: understanding the structure of bases or common bases of matroids. These conjectures suggest that matroids may possess much stronger structural properties than are…
Attribute reduction is a basic issue in knowledge representation and data mining. Rough sets provide a theoretical foundation for the issue. Matroids generalized from matrices have been widely used in many fields, particularly greedy…
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and…
We characterize the shifted simple graphs and the $3$-uniform shifted hypergraphs whose inverse image under exterior shifting is the set of bases of a matroid: those are exactly the hypergraphs whose hyperedges form an initial lex-segment.…
A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a…
Conditional independence has been widely used in AI, causal inference, machine learning, and statistics. We introduce categoroids, an algebraic structure for characterizing universal properties of conditional independence. Categoroids are…
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…
This thesis is basically devoted to matroids -- fundamental structure of combinatorial optimization -- though some of our results concern simplicial complexes, or Euclidean spaces. We study old and new problems for these structures, with…
The Whitney module of a matroid is a natural analogue of the tensor algebra of the exterior algebra of a vector space that takes into account the dependencies of a matroid. In this paper we indicate the role that tableaux can play in…
In this note we characterize tropical bases as sets of circuits that by orthogonality determine the set of cocircuits of a simple matroid. Furthermore, we show that any circuit, which itself is closed, must be contained in any tropical…
We present exponential and super factorial lower bounds on the number of Hamiltonian cycles passing through any edge of the basis graphs of a graphic, generalized Catalan and uniform matroids. All lower bounds were obtained by a common…
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…
Rough sets are efficient for data pre-processing in data mining. Matroids are based on linear algebra and graph theory, and have a variety of applications in many fields. Both rough sets and matroids are closely related to lattices. For a…
OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number…
Rough sets are efficient for data pre-processing in data mining. As a generalization of the linear independence in vector spaces, matroids provide well-established platforms for greedy algorithms. In this paper, we apply rough sets to…
We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of…
In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related…
A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lov\'asz initiated the study of matroids from…