Pair Of Matroids Research Articles

Connectivity gaps among matroids with the same enumerative invariants

Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the G-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer n, there are pairs of matroids that have the same configuration (and so the same G-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds n, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.

The Free m-Cone of a Matroid and Its $${\mathcal {G}}$$-Invariant

For a matroid, its configuration determines its \({\mathcal {G}}\)-invariant. Few examples are known of pairs of matroids with the same \({\mathcal {G}}\)-invariant but different configurations. In order to produce new examples, we introduce the free m-cone \(Q_m(M)\) of a loopless matroid M, where m is a positive integer. We show that the \({\mathcal {G}}\)-invariant of M determines the \({\mathcal {G}}\)-invariant of \(Q_m(M)\), and that the configuration of \(Q_m(M)\) determines M; so if M and N are nonisomorphic and have the same \({\mathcal {G}}\)-invariant, then \(Q_m(M)\) and \(Q_m(N)\) have the same \({\mathcal {G}}\)-invariant but different configurations. We prove analogous results for several variants of the free m-cone. We also define a new matroid invariant of M, and show that it determines the Tutte polynomial of \(Q_m(M)\).

Dual Pairs of Matroids with Coefficients in Finitary Fuzzy Rings of Arbitrary Rank

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

Tutte’s dichromate for signed graphs

We introduce the trivariate Tutte polynomial of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the “dichromate” for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank–size generating function).The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set — for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.

The Almost Intersection Property for Pairs of Matroids on Common Groundset

We introduce the Almost Intersection Property for pairs of possibly infinite matroids on the same groundset. We prove that if such a pair satisfies the Almost Intersection Property then it satisfies the Matroid Intersection Conjecture of Nash-Williams. We also present some corollaries of that result.

Cyclic flats and corners of the linking polynomial

Abstract Let T(M; x,y) = ∑ij Tij xiyjdenote the Tutte polynomial of the matroid M. If Tij is a corner of T (M; x, y), then Tij counts the sets of corank i and nullity j and each such set is a cyclic flat of M. The main result of this article consists of extending the definition of cyclic flats to a pair of matroids and proving that the corners of the linking polynomial give the lower bound of the number of the cyclic flats of the matroid pair.

On chains of matroids in the weak order

Weak maps in matroids originate an order called weak order. In this paper, we characterize some chains of matroids which cannot be increased with respect to the weak order. We also give conditions for a pair of matroids to have few non-common bases.

The edge covering number of the intersection of two matroids

The edge covering number of a hypergraph A is β(A)=min{|B|:B⊆A,⋃B=⋃A}. The paper studies a conjecture on the edge covering number of the intersection of two matroids. For two natural numbers k,ℓ, let f(k,ℓ) be the maximal value of β(M∩N) over all pairs of matroids M,N such that β(M)=k and β(N)=ℓ. In (Aharoni and Berger, 2006) [1] the first two authors proved that f(k,ℓ)≤2max(k,ℓ) and conjectured that f(k,k)=k+1 and f(k,ℓ)=ℓ when ℓ>k. In this paper we prove that f(k,k)≥k+1, f(2,2)=3 and f(2,3)≤4. We also form a conjecture on the edge covering number of 2-polymatroids that is a common extension of the above conjecture and the Goldberg–Seymour conjecture, and prove its first non-trivial case.

Splicing matroids

We introduce and study a natural variant of matroid amalgams. For matroids M ( A ) and N ( B ) with M . ( A ∩ B ) = N | ( A ∩ B ) , we define a splice of M and N to be a matroid L on A ∪ B with L | A = M and L . B = N . We show that splices exist for each such pair of matroids M and N ; furthermore, there is a freest splice of M and N , which we call the free splice. We characterize when a matroid L ( A ∪ B ) is the free splice of L | A and L . B . We study minors of free splices and the interaction between free splice and several other matroid operations. Although free splice is not an associative operation, we prove a weakened counterpart of associativity that holds in general and we characterize the triples for which associativity holds. We also study free splice as it relates to various classes of matroids.

Parametric nonlinear discrete optimization over well-described sets and matroid intersections

We address optimization of parametric nonlinear functions of the form f(Wx), where $${f : \mathbb {R}^d \rightarrow \mathbb {R}}$$is a nonlinear function, W is a d × n matrix, and feasible x are in some large finite set $${\mathcal {F}}$$of integer points in $${\mathbb {R}^n}$$. Generally, such problems are intractable, so we obtain positive algorithmic results by looking at broad natural classes of f, W and $${\mathcal {F}}$$. One of our main motivations is multi-objective discrete optimization, where f trades off the linear functions given by the rows of W. Another motivation is that we want to extend as much as possible the known results about polynomial-time linear optimization over trees, assignments, matroids, polymatroids, etc. to nonlinear optimization over such structures. We assume that $${\mathcal {F}}$$is well described (i.e., we can efficiently optimize a linear objective function on $${\mathcal {F}}$$; equivalently, we have an efficient separation oracle for the convex hull of $${\mathcal {F}}$$). For example, the sets of characteristic vectors of (i) matchings of a graph, and (ii) common bases of a pair of matroids on a common ground set satisfy this property. In this setting, the problem is already known to be intractable (even for a single matroid), for general f (given by a comparison oracle), for (i) d = 1 and binary-encoded W, and for (ii) d = n and W = I. Our main results (a few technicalities and some generality suppressed): When $${\mathcal {F}}$$is well described, f is convex (or even quasi-convex), and W has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic algorithm for maximization. When $${\mathcal {F}}$$is well described, f is a norm, and W is binary-encoded, we give an efficient deterministic constant-approximation algorithm for maximization (Note that the approximation factor depends on the norm, and hence implicitly on the number of rows of W, while the running time increases only linearly in the number of rows of W). When non-negative $${\mathcal {F}}$$is well described, f is “ray concave” and non-decreasing, and non-negative W has a fixed number of rows and is unary encoded or with entries in a fixed set, we give an efficient deterministic constant-approximation algorithm for minimization. When $${\mathcal {F}}$$is the set of characteristic vectors of common independent sets or bases of a pair of rational vectorial matroids on a common ground set, f is arbitrary, and W has a fixed number of rows and is unary encoded, we give an efficient randomized algorithm for optimization.

A Construction of Infinite Sets of Intertwines for Pairs of Matroids

An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other by taking free extensions, free coextensions, and minors, we construct a family of rank-k intertwines for each sufficiently large integer k. We also treat some properties of these intertwines.

INFINITE MATROIDAL VERSION OF HALL'S MATCHING THEOREM

Hall's theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv have generalized the notion of matchability to a pair of possibly infinite matroids on the same set and given a condition that is sufficient for the matchability of a given pair (M, W) of finitary matroids, where the matroid M is SCF (a sum of countably many matroids of finite rank). The condition of Aharoni and Ziv is not necessary for matchability. The paper gives a condition that is necessary for the existence of a matching for any pair of matroids (not necessarily finitary) and is sufficient for any pair (M, W) of finitary matroids, where the matroid M is SCF.

The free product of matroids

We introduce a noncommutative binary operation on matroids, called free product. We show that this operation respects matroid duality, and has the property that, given only the cardinalities, an ordered pair of matroids may be recovered, up to isomorphism, from its free product. We use these results to give a short proof of Welsh’s 1969 conjecture, which provides a progressive lower bound for the number of isomorphism classes of matroids on an n-element set.

A linking polynomial of two matroids

We introduce a 4-variable polynomial Q as a natural invariant linking a pair of matroids on a common ground set. The polynomial Q has similarities with the classical Tutte polynomial of a single matroid, it contains as specialisations the generating function of common independent and spanning sets of a given size, it behaves naturally under a duality transform and there is a recipe theorem which shows that essentially it is the unique invariant satisfying simultaneous delete/contract recursions on a pair of matroids.

On matroid intersection adjacency

This paper is devoted to a simple alternative proof for a theorem of Frank and Tardos (Math. Programming 42 (1988) 489) on adjacency of extreme points in the common base polytope of a pair of matroids. The new proof relies merely on the simultaneous base exchange axiom, independent of the linear inequality description of the polytope due to Edmonds (Guy et al. (Eds.), Combinatorial Structures and Their Applications, Gordon and Breach, London, 1970, p. 69).

A dual approximation approach to weighted matroid intersection

This paper presents a simple algorithm for finding a minimum weight common base of a pair of matroids. The algorithm approximately splits the weight and maintains two optimal bases with respect to the split weights. The algorithm consists of only local operations and is fairly easy to implement.

An extension of elementary bipartite graphs to matroid pairs and its characterization

In this paper we shall present a generalization of elementary bipartite graphs to a certain pair of matroids, which we shall call an “elementary pair”, and show that a matroid pair is elementary if and only if it is irreducible with respect to a decomposition method associated with matroid intersection problems.

A Characterization of Tutte Invariants of 2-Polymatroids

This paper develops a theory of Tutte invariants for 2-polymatroids that parallels the corresponding theory for matroids. It is shown that such 2-polymatroid Invariants arise in the enumeration of a wide variety of combinatorial structures including matchings and perfect matchings in graphs, weak colourings in hypergraphs, and common bases in pairs of matroids. The main result characterizes all such invariants proving that, with some trivial exceptions, every 2-polymatroid Tutte invariant can be easily expressed in terms of a certain two-variable polynomial that is closely related to the Tutte polynomial of a matroid.

Blocking, antiblocking, and pairs of matroids and polymatroids

I extend two theorems of Edmonds concerning common independent vectors in two polymatroids. These results give a proof of conjecture of Fulkerson on the blocker associated with the collection of k-sets independent in two matroids.

Patroids

A matroid M over a set E of elements is semiseparated by a partition { S 1, S 2} of E iff rank E = rank S 1 + rank S 2 + 1. Such a semiseparation defines in each S i a pair of matroids or patroid P i = ( M i , m i ); the two patroids P 1, P 2 weld to form M. The operations of removing and contracting a non-degenerate element of a matroid produce a patroid. The properties of patroids, their bases, and circuits are discussed.