D. Molodtsov (1999) introduced the concept of a soft set as a new approach for modeling uncertainties. The aim of this work is to define special kinds of soft sets, namely soft, L-fuzzifying soft, L-soft, and L-fuzzy soft neighborhood sets and to use them in order to give an alternative characterization of categories related to topology: crisp topological, L-topological, L-fuzzifying topologica...

This article introduces the lazy matroid problem, which captures the goal of saving time or money in certain task selection scenarios. We are given a budget B and a matroid M with weights on its elements. The problem consists in finding an independent set F of minimum weight. In addition, F is feasible if its augmentation with any new element x implies that either F + x exceeds B or F + x is de...

Many problems that can be correctly solved by greedy algorithms can be described in terms of an abstract combinatorial object called a matroid. Matroids were first described in 1935 by the mathematician Hassler Whitney as a combinatorial generalization of linear independence of vectors—‘matroid’ means ‘something sort of like a matrix’. A matroid M is a finite collection of finite sets that sati...

The multivariate Tutte polynomial ẐM of a matroid M is a generalization of the standard two-variable version, obtained by assigning a separate variable ve to each element e of the ground set E. It encodes the full structure of M . Let v = {ve}e∈E , let K be an arbitrary field, and suppose M is connected. We show that ẐM is irreducible over K(v), and give three self-contained proofs that the Gal...

We express the matroid polytope PM of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of PM . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Grk,n. We then derive analogous results for the independent set polytope and the associated flag matroid polytope of M . Our proofs are based on a natural...

A matroid M is sequential or has path width 3 if M is 3-connected and its ground set has a sequential ordering, that is, an ordering (e1, e2, . . . , en) such that ({e1, e2, . . . , ek}, {ek+1, ek+2, . . . , en}) is a 3-separation for all k in {3, 4, . . . , n − 3}. This paper proves that every sequential matroid is easily constructible from a uniform matroid of rank or corank two by a sequence...

Many problems that can be correctly solved by greedy algorithms can be described in terms of an abstract combinatorial object called a matroid. Matroids were first described in 1935 by the mathematician Hassler Whitney as a combinatorial generalization of linear independence of vectors—‘matroid’ means ‘something sort of like a matrix’. A matroid M is a finite collection of finite sets that sati...

A symmetric matroid is a matroid deened on the edge-set of some countably innnite complete graph K in a way that ranks of nite sub-graphs of K are invariant under isomorphism. Thus a symmetric ma-troid M induces on any nite graph G a uniquely determined matroid M(G). We study connectivity properties of circuits and generalized trees of symmetric matroids. We give several characterizations of a ...

This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M . We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank 3 oriented matroid with up to eight points is contractible. For these s...