This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function mu(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, mu(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether mu(G)<= k. Using combinatorial arguments, we prove that mu(n)is in Omega(loglog n). On the other hand, we establish that mu(n) is in O(log n / loglog n). Finally, we prove that mu(n)=4 for n=5,...,12, and mu(n)=5 for n=13,14,15.
Matroid is a generalization of many fundamental objects in combinatorial mathematics , and matroid intersection problem is a classical subject in combinatorial optimization . However , only the intersection of two matroids are well understood . The solution of the intersection problem of more than three matroids is proved to be \textbf{NP-hard} . We will give a lower bound estimate on the maximal cardinality of the common independent sets in matroid intersections . We will also study some properties of the intersection of more than two matroids and deduce some analogous results for Edmonds' Mi…
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most $48$ elements.
We construct some matroids that have a circuit and a cocircuit with infinite intersection. This answers a question of Bruhn, Diestel, Kriesell, Pendavingh and Wollan. It further shows that the axiom system for matroids proposed by Dress does not axiomatize all infinite matroids. We show that one of the matroids we define is a thin sums matroid whose dual isn't a thin sums matroid.
We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved recently by Aharoni and Berger. 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.
Let $ E $ be a possibly infinite set and let $ M $ and $ N $ be matroids defined on $ E $. We say that the pair $ \{ M,N \} $ has the Intersection property if $ M $ and $ N $ share an independent set $ I $ admitting a bipartition $ I_M\sqcup I_N $ such that $ \mathsf{span}_M(I_M)\cup \mathsf{span}_N(I_N)=E $. The Matroid Intersection Conjecture of Nash-Williams says that every matroid pair has the Intersection property. The conjecture is known and easy to prove in the case when one of the matroids is uniform and it was shown by Bowler and Carmesin that the conjecture is implied by its special …
As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and…
We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d = r \lceil k \rceil - r + 1$ functions of size (=sum of values) $k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i \leq d$, supporting a function with the same properties.
We prove that if $ M $ and $ N $ are finitary matroids on a common countable edge set $ E $ then they admit a common independent set $I $ such that there is a bipartition $ E=E_{M}\cup E_{N} $ for which $ I\cap E_M $ spans $ E_M $ in $ M $ and $ I\cap E_N $ spans $ E_N $ in $ N $. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.
Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts.
Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two examples, Webb (2004) introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy-Binet formula. This paper studies counting on linear matroid problems extending Webb's work. We first introduce "Pfaffian parities" as an extension of Pfaffian pairs to the linear matroid parity problem, which is a commo…
Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the ra…
In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special 16-element matroid $Y_{16}$, a 3-connected regular matroid, a binary spike with rank at least four, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid $M^*(K_{3, n})$, $M^*(K_{3, n}^{\prime})$, $M^*(K_{3, n}^{\prime\prime})$, or $M^*(K_{3, n}^{\prime\prime\prime})$ ($n\ge 2$). Here the simple graphs $K_{3, n}^{\prime}, K_…
In this paper, we investigate the classes of matroid intersection admitting a solution for the problem of partitioning the ground set $E$ into $k$ common independent sets, where $E$ can be partitioned into $k$ independent sets in each of the two matroids. For this problem, we present a new approach building upon the generalized-polymatroid intersection theorem. We exhibit that this approach offers alternative proofs and unified understandings of previous results showing that the problem has a solution for the intersection of two laminar matroids and that of two matroids without $(k+1)$-spanned…
We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin. This product generalises the product of Allermann and Rau, and Allermann and also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one dimensional fan cycles in two dimensional matroidal fans and given an application of th…
For two matroids $M_1$ and $M_2$ with the same ground set $V$ and two cost functions $w_1$ and $w_2$ on $2^V$, we consider the problem of finding bases $X_1$ of $M_1$ and $X_2$ of $M_2$ minimizing $w_1(X_1)+w_2(X_2)$ subject to a certain cardinality constraint on their intersection $X_1 \cap X_2$. For this problem, Lendl, Peis, and Timmermans (2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $|X_1 \cap X_2|\le k$ or $|X_1 \cap X_2|\ge k$; and designed a new primal-dual algorithm for the case where…
For a simplicial complex ${\mathcal C}$ denote by $β({\mathcal C})$ the minimal number of edges from ${\mathcal C}$ needed to cover the ground set. If ${\mathcal C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in {\mathcal C}$ meeting each $A_i$ in at least $\frac{|A_i|}{β({\mathcal C})}$ elements. We conjecture that a slightly weaker result is true for the intersections of two matroids: if ${\mathcal D}={\mathcal P} \cap {\mathcal Q}$, where ${\mathcal P},{\mathcal Q}$ are matroids on the same ground set $V$ and $β({\mathcal P}), β({\mathc…
We study a particular graded ring structure on the set of all loopfree matroids on a fixed labeled ground set, which occurs naturally in tropical geometry. The product is given by matroid intersection and the additive structure is defined by assigning to each matroid the indicator vector of its chains of flats. We show that this ring is generated in corank one, more precisely that any matroid can be written as a linear combination of products of corank one matroids. Moreover, we prove that a basis for the graded part of rank r matroids is given by the set of nested matroids and that the total …
Given sets $F_1, \ldots ,F_n$, a {\em partial rainbow function} is a partial choice function of the sets $F_i$. A {\em partial rainbow set} is the range of a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if $M$ and $N$ are matroids on the same ground set, and $F_1, \ldots ,F_n$ are pairwise disjoint sets of size $n$ belonging to $M \cap N$, then there exists a rainbow set of size $n-1$ belonging to $M \cap N$. Following an idea of Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a rainbow set of size at least $n-\sqrt{n}$.
In this paper, we investigate three problems concerning the toric ideal associated to a matroid. Firstly, we list all matroids $\mathcal M$ such that its corresponding toric ideal $I_{\mathcal M}$ is a complete intersection. Secondly, we handle with the problem of detecting minors of a matroid $\mathcal M$ from a minimal set of binomial generators of $I_{\mathcal M}$. In particular, given a minimal set of binomial generators of $I_{\mathcal M}$ we provide a necessary condition for $\mathcal M$ to have a minor isomorphic to $\mathcal U_{d,2d}$ for $d \geq 2$. This condition is proved to be suff…
Given a mixed hypergraph $\mathcal{F}=(V,\mathcal{A}\cup \mathcal{E})$, functions $f,g:V\rightarrow \mathbb{Z}_+$ and an integer $k$, a packing of $k$ spanning mixed hyperarborescences is called $(k,f,g)$-flexible if every $v \in V$ is the root of at least $f(v)$ and at most $g(v)$ of the mixed hyperarborescences. We give a characterization of the mixed hypergraphs admitting such packings. This generalizes results of Frank and, more recently, Gao and Yang. Our approach is based on matroid intersection, generalizing a construction of Edmonds. We also obtain an algorithm for finding a minimum we…
We address optimization of nonlinear functions of the form $f(Wx)$, where $f:\R^d\to \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $F$ of integer points in $\R^n$. One motivation is multi-objective discrete optimization, where $f$ trades off the linear functions given by the rows of $W$. Another motivation is to extend known results about polynomial-time linear optimization over discrete structures to nonlinear optimization. We assume that the convex hull of $F$ is well-described by linear inequalities. For example, the set of character…
The Packing/Covering Conjecture was introduced by Bowler and Carmesin motivated by the Matroid Partition Theorem by Edmonds and Fulkerson. A packing for a family $ (M_i: i\inΘ) $ of matroids on the common edge set $ E $ is a system $ (S_i: i\inΘ) $ of pairwise disjoint subsets of $ E $ where $ S_i $ is panning in $ M_i $. Similarly, a covering is a system $ (I_i: i\inΘ) $ with $\bigcup_{i\inΘ} I_i=E $ where $ I_i $ is independent in $ M_i $. The conjecture states that for every matroid family on $ E $ there is a partition $E=E_p \sqcup E_c$ such that $ (M_i \upharpoonright E_p: i\in Θ) $ admit…
We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is e…
We characterize the 3-connected members of the intersection of the class of bicircular and cobicircular matroids. Aside from some exceptional matroids with rank and corank at most 5, this class consists of just the free swirls and their minors.
This is a survey article on the exchange properties characterizing M-natural-concave set functions and valuated matroids (M-concave set functions). The objective of this paper is to collect related results scattered in the literature and to give (reasonably) self-contained elementary proofs for them.
Given an $\Bbb{F}$-represented matroid $(M,ρ)$ with the ground set $[m]$, the representation $ρ$ naturally defines a hyperplane arrangement $\mathcal{A}_ρ$. We will study its parallel translates $\mathcal{A}_{ρ,{g}}$ of $\mathcal{A}_ρ$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{ρ,{ g}})$ and the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{δρ}$, which is a hyperplane arrangement associated with the derived matroid $(δM,δρ)$ and also known as the discriminantal a…
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture.
The Lagrangian geometry of matroids was introduced in [ADH20] through the construction of the conormal fan of a matroid M. We used the conormal fan to give a Lagrangian-geometric interpretation of the h-vector of the broken circuit complex of M: its entries are the degrees of the mixed intersections of certain convex piecewise linear functions $γ$ and $δ$ on the conormal fan of M. By showing that the conormal fan satisfies the Hodge-Riemann relations, we proved Brylawski's conjecture that this h-vector is a log-concave sequence. This sequel explores the Lagrangian combinatorics of matroids, fu…
Oxley conjectured (1992) that if a matroid has a circuit-cocircuit intersection of size $k\ge4$, it has a circuit-cocircuit intersection of size $k-2$. We show that this conjecture holds for $k\le6$.
An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable.
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson.
Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present general upper and lower bounds in the setting of general connected orientable matroids, leading to the study of subgraphs of the base graph and the intersection graph of circuits. We then consider the problem for uniform matroids which is closely related to the notion of (connected) covering numbers in Design Theory. Finally, we also devote special attentio…
We consider the class of {\em separable} $k$-hypergraphs, which can be viewed as uniform analogs of threshold Boolean functions, and the class of {\em equatable} $k$-hypergraphs. We show that every $k$-hypergraph is either separable or equatable but not both. We raise several questions asking which classes of equatable (and separable) hypergraphs enjoy certain appealing characterizing properties, which can be viewed as uniform analogs of the $2$-summable and $2$-monotone Boolean function properties. In particular, we introduce the property of {\em exchangeability}, and show that all these ques…
It is proved that the broken circuit complex of an ordered matroid is Gorenstein if and only if it is a complete intersection. Several characterizations for a matroid that admits such an order are then given, with particular interest in the $h$-vector of broken circuit complexes of the matroid. As an application, we prove that the Orlik--Terao algebra of a hyperplane arrangement is Gorenstein if and only if it is a complete intersection. Interestingly, our result shows that the complete intersection property (and hence the Gorensteinness as well) of the Orlik--Terao algebra can be determined f…
Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a chord diagram depends on the intersection graph of the diagram rather than on the diagram itself. Moreover, it was shown that the value of this weight system on an intersection graph depends on the cy- cle matroid of the graph rather than on the graph itself. This result arose the question whether there is a natural way to introduce a 4-term relation on the sp…
