Two algorithms for matroids

A matroid may be characterized by the collection of its bases or by the collection of its circuits. Along with any matroid is a uniquely determined dual matroid. Given the bases of a matroid, it is possible to show that base complements are precisely the bases of the dual. So it is easy to construct bases of the dual given the bases of the original matroid.This paper provides two results. The first enables construction of all circuits of the dual matroid from circuits of the original matroid. The second constructs all bases of a matroid from its circuits.     

An Adjacency Criterion for Coxeter Matroids

A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups A _n , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope. In particular, a criterion is given for adjacency of vertices in the matroid polytope.     

On the (co)girth of a connected matroid

This article studies the girth and cogirth problems for a connected matroid. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a vector matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose contraction or restriction is disconnected. Then, we devise algorithms that find the cogirth of a matroid M from the matroids associated with the direct sum components of the restriction of M. As a result, the problem of finding the (co)girth of a matroid can be decomposed into a set of smaller sub-problems, which helps alleviate the computation. Finally, we implement and demonstrate the application of our algorithms to vector matroids.     

Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.     

Flag enumerations of matroid base polytopes

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.     

双圈拟阵

Sim■es Pereira于1992年提出双圈拟阵.本文讨论了(i)双圈拟阵及其秩函数;(ii)次模函数在双圈拟阵中的应用;(iii)双圈拟阵B(G)的横贯拟阵.主要结果:1°由圈矩阵Bf=[I,Bf12]和圈秩的概念,推出M(f0)为双圈拟阵;2°证明了双圈拟阵B(G)等于由子集族{Av∶v∈V(G)},e与v在G中相关联}所确定的横贯拟阵;3°用不同于Matthews(1977)的方法证明了(iii).     

拟阵的可去圈

本首先用拟阵语言将图论的新概念定义成了拟阵的新概念，然后用拟阵语言将Goddyn和Heuevl所得的图论上的新结果平移成了拟阵的新结果，最后用拟阵的方法对它们给出了新的证明。     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

Decomposition of binary matroids

We prove results relating to the decomposition of a binary matroid, including its uniqueness when the matroid is cosimple. We extend the idea of “freedom” of an element in a matroid to “freedom” of a set, and show that there is a unique maximal integer polymatroid inducing a given binary matroid.     

Truncation Formulas for Invariant Polynomials of Matroids and Geometric Lattices

This paper considers the truncation of matroids and geometric lattices. It is shown that the truncated matroid of a representable matroid is again representable. Truncation formulas are given for the coboundary and M?bius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the truncation formula of the rank generating polynomial of a matroid by Britz.     

Extremal critically connected matroids

A connected matroid M is called a critically connected matroid if the deletion of any one element from M results in a disconnected matroid. We show that a critically connected matroid of rank n, n≥3, can have at most 2n?2 elements. We also show that a critically connected matroid of rank n on 2n?2 elements is isomorphic to the forest matroid of K₂, n?2.     

Coxeter matroid polytopes

If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge. The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialize this information to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical matroid polytope familiar to matroid theorists. This work was supported in part by NSA grant MDA904-95-1-1056.     

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.     

The circuit basis in binary matroids

The concept of a circuit basis for a matroid is introduced, as an algorithmically rapid way of determining several characteristics of a given matroid. It is used to give a short search for planarity in graphs, and also to begin the answer to a question of G.-C. Rota about “dependency among dependencies.” A circuit basis for a matroid is a least set of circuits which will generate all the circuits of the matroid by repeated use of symmetric differences of cells.     

Quasi‐graphic matroids*

Frame matroids and lifted‐graphic matroids are two interesting generalizations of graphic matroids. Here, we introduce a new generalization, quasi‐graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted‐graphic matroids, it is easy to certify that a 3‐connected matroid is quasi‐graphic. The main result is that every 3‐connected representable quasi‐graphic matroid is either a lifted‐graphic matroid or a frame matroid.     

A Note on the Sticky Matroid Conjecture

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzík proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r ≥ 3, no rank−r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.     

准模糊图拟阵基图的最大权基与字典序最大的基

模糊拟阵的基图是模糊拟阵的基本概念.在准模糊图拟阵的基础上,给出了准模糊图拟阵基图的最大权基与字典序最大的基的性质,这将有利于模糊拟阵从基础研究逐渐转向应用研究.     

Vámos拟阵的直立

通过利用一个拟阵与它的直立之间的关系,特别是通过处理一个拟阵与一个特殊映射f之间的关系,具体分析了Vámos拟阵的直立问题,得到了除去平凡情形,Vámos拟阵没有直立的结论.从而回答了由Welsh提出"Vámos拟阵是否有直立?"的问题.     

拟阵与概念格的关系

本文以构造的方式建立起拟阵与概念格的联系,得到在同构意义下每个拟阵是一个概念格,但反之不然的结论;该结论使得利用概念格的性质研究拟阵成为现实,特别为将建造概念格的算法尤其是已计算机化的算法应用于求取拟阵奠定了基础,也为拟阵论成为研究概念格性质的辅助工具打下基础．     

Separating cocircuits in binary matroids

A cocircuit of a matroid is separating if deleting it leaves a separable matroid. We give an effecient algorithm which finds a separating cocircuit or a Fano minor in a binary matroid, thus proving constructively a theorem of Tutte. Using this algorithm and a new recursive characterization of bond matroids, we give a new method for testing binary matroids for graphicness. We also give an efficient algorithm for finding a special kind of separating cocircuit: one whose deletion leaves a matroid having a coloop.