The Tchebyshev Transforms of the First and Second Kind

An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge’s peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg’s classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg’s r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.     

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

Recipe theorem for the Tutte polynomial for matroids,renormalization group-like approach

Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in [W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999) 297–300] and [G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, Discrete Math. 179 (1998) 111–119].     

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.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

Ideal secret sharing schemes whose minimal qualified subsets have at most three participants

One of the main open problems in secret sharing is the characterization of the access structures of ideal secret sharing schemes. Brickell and Davenport proved that every one of these ideal access structures is related in a certain way to a unique matroid. Specifically, they are matroid ports. In addition to the search of general results, this difficult open problem has been studied in previous works for several families of access structures. In this paper we do the same for access structures with rank 3, that is, structures whose minimal qualified subsets have at most three participants. We completely characterize and classify the rank-3 access structures that are matroid ports. We prove that all access structures with rank three that are ports of matroids greater than 3 are ideal. After the results in this paper, the only open problem in the characterization of the ideal access structures with rank three is to characterize the rank-3 matroids that can be represented by an ideal secret sharing scheme. A previous version of this paper appeared in Fifth Conference on Security and Cryptography for Networks, SCN 2006, Lecture Notes in Computer Science 4116 (2006) 201–215.     

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K ₅\ e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F ₇ with itself across a triangle with an element of the triangle deleted; it’s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in P ₉. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F ₇ and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R ₁₀, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012).     

Extension spaces of oriented matroids

We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the “Generalized Baues Problem” of Billera, Kapranov, and Sturmfels, via the Bohne-Dress theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not known whether the extension space is connected for all realizable oriented matroids (hyperplane arrangements). We show that the subspace of realizable extensions is always connected but not necessarily spherical. Nonrealizable oriented matroids of rank 4 with disconnected extension spaces were recently constructed by Mnëv and Richter-Gebert.     

Persistency in combinatorial optimization problems on matroids

Optimization problems on matroids are generalizations of such important combinatorial optimization problems like the problem of minimum spanning tree of a graph, the bipartite matching problem, flow problems, etc. We analyze algorithms for finding the maximum weight independent set of a matroid and for finding a maximum cardinality intersection of two matroids and extend them to obtain the so-called “persistency” partition of the basic set of the matroid, where contain elements belonging to all optimum solutions; contain elements not belonging to any optimum solution; contain elements that belong to some but not to all optimum solutions.     

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids M and N is maximal with respect to the weak order among matroids having M as a submatroid, with complementary contraction equal to N. Any minor of the free product of M and N is a free product of a repeated truncation of the corresponding minor of M with a repeated Higgs lift of the corresponding minor of N. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra of a family of matroids that is closed under formation of minors and free products: namely, is cofree, cogenerated by the set of irreducible matroids belonging to .     

Unipancyclic matroids

Analogous to the concept of uniquely pancyclic graphs, we define a uniquely pancyclic (UPC) matroid of rank $r$ to be a (simple) rank-$r$ matroid containing exactly one circuit of each length $?$ for $3\le ?\le r+1$. Our discussion addresses the existence of graphic, binary, and transversal representations of UPC matroids. Using Shi’s results, which catalogued exactly seven non-isomorphic UPC graphs, we produce a nongraphic binary UPC matroid of rank 24. We consider properties of binary UPC matroids in general, and prove that all binary UPC matroids have a connectivity of $2$.     

A constrained independent set problem for matroids

In this note, we study a constrained independent set problem for matroids. The problem can be regarded as an ordered version of the matroid parity problem. By a reduction of this problem to matroid intersection, we prove a min-max formula. We show how earlier results of Hefner and Kleinschmidt on the so-called MS-matchings fit in our framework.     

Matroids with few non-common bases

In [On Mills's conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271-276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195-205]: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂≤k. In [Matroids with many common bases, Discrete Math. 270 (2003) 193-205], Lemos proved a similar result, where the hypothesis of the matroids being k-connected is replaced by the weaker hypothesis of being vertically k-connected. In this paper, we extend these results.     

A composition for matroids

The composition of a quotient matroid Q over a collection of component matroids f₁, …, f_n indexed on the cells of Q, is described. This composition, called quotient composition, may be viewed as an application of clutter composition to matroids, or as a generalization of matroid direct sum composition to the next higher connectivity. It may also be viewed as equivalent to compositions described by Minty in 1966, and Brylawski in 1971.Quotient composition is characterized, and the circuits and rank function of a composed matroid are given. Various other properties are described, along with a category for quotient composition.     

On finite matroids with two more hyperplanes than points

One of the most interesting results about finite matroids of finite rank and generalized projective spaces is the result of Basterfield, Kelly and Green (1968/1970) (J.G. Basterfield, L.M. Kelly, A characterization of sets of n points which determine n hyperplanes, in: Proceedings of the Cambridge Philosophical Society, vol. 64, 1968, pp. 585-588; C. Greene, A rank inequality for finite geometric lattices, J. Combin Theory 9 (1970) 357-364) affirming that any matroid contains at least as many hyperplanes as points, with equality in the case of generalized projective spaces. Consequently, the goal is to characterize and classify all matroids containing more hyperplanes than points. In 1996, I obtained the classification of all finite matroids containing one more hyperplane than points. In this paper a complete classification of finite matroids with two more hyperplanes than points is obtained. Moreover, a partial contribution to the classification of those matroids containing a certain number of hyperplanes more than points is presented.     

Systems of Polynomials for Detecting Orientable Matroids

We study systems of polynomial equations that correspond to a matroid M. Each of these systems has a zero solution if and only if M is orientable. Since determining if a matroid is orientable is NP-complete with respect to the size of the input data, determining if these systems have solutions is also NP-complete. However, we show that one of the associated polynomial systems corresponding to M is linear if M is a binary matroid and thus it may be determined if binary matroids are orientable in polynomial time given the circuits and cocircuits of said matroid as the input. In the case when M is not binary, we consider the associated system of non-linear polynomials. In this case Hilbertʼs Nullstellensatz gives us that M is non-orientable if and only if a certain certificate to the given polynomials system exists. We wish to place bounds on the degree of these certificates in future research.     

Symmetric and quasi-symmetric functions associated to polymatroids

To every subspace arrangement X we will associate symmetric functions ℘[X] and ℋ[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant ℋ[X] specializes to the Tutte polynomial . Billera, Jia and Reiner recently introduced a quasi-symmetric function ℱ[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function for polymatroids which has this property as well. Moreover, specializes to ℘[X], ℋ[X], and ℱ[X]. The author is partially supported by the NSF, grant DMS 0349019.     

A matroid operation that commutes with duality

In this note, we propose an operation for matroids that commutes with duality having deletions and contractions as extremal cases. Crapo and Schmitt's free product of matroids is one of its consequences. A special case of this operation can be used as an inductive tool because it reduces the number of elements of a matroid by two and it is invariant by duality.     

Random sampling and greedy sparsification for matroid optimization problems

Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Application of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by agreedy algorithm that grows an independent set into an optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a givenfixed basis is optimum, showing that the two problems can be solved in roughly the same time. Another application of sampling is to packing matroid bases, also known as matroid partitioning. Sampling reduces the number of bases that must be packed. We combine sampling with a greedy packing strategy that reduces the size of the matroid. Together, these techniques give accelerated packing algorithms. We give particular attention to the problem of packing spanning trees in graphs, which has applications in network reliability analysis. Our results can be seen as generalizing certain results from random graph theory. The techniques have also been effective for other packing problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Some of this work done at Stanford University, supported by National Science Foundation and Hertz Foundation Graduate Fellowships, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation and Xerox Corporation. Also supported by NSF award 962-4239.