Matroid

Free matroid Let E be a finite set. The set of all subsets of E defines the independent sets of a matroid. It is called the free matroid over E. Uniform matroids Let E be a finite set and k a natural number. One may define a matroid on E by taking every subset of E to be a basis. This is known as the uniform matroid of rank k. A uniform matroid with rank k and with n elements is denoted U_{k,n}. All uniform matroids of rank at least 2 are simple (see ). The uniform matroid of rank 2 on n points is called the n point line. A matroid is uniform if and only if it has no circuits of size less than one plus the rank of the matroid. The direct sums of uniform matroids are called partition matroids. In the uniform matroid U_{0,n}, every element is a loop (an element that does not belong to any independent set), and in the uniform matroid U_{n,n}, every element is a coloop (an element that belongs to all bases). The direct sum of matroids of these two types is a partition matroid in which every element is a loop or a coloop; it is called a discrete matroid. An equivalent definition of a discrete matroid is a matroid in which every proper, non-empty subset of the ground set E is a separator. Matroids from linear algebra . It is GF(2)-linear but not real-linear.|class=skin-invert-image , not linear over any field|class=skin-invert-image Matroid theory developed mainly out of a deep examination of the properties of independence and dimension in vector spaces. There are two ways to present the matroids defined in this way: : If E is any finite subset of a vector space V, then we can define a matroid M on E by taking the independent sets of M to be the linearly independent subsets of E. The validity of the independent set axioms for this matroid follows from the Steinitz exchange lemma. : If M is a matroid that can be defined in this way, we say the set E represents M. : Matroids of this kind are called vector matroids. An important example of a matroid defined in this way is the Fano matroid, a rank three matroid derived from the Fano plane, a finite geometry with seven points (the seven elements of the matroid) and seven lines (the proper nontrivial flats of the matroid). It is a linear matroid whose elements may be described as the seven nonzero points in a three dimensional vector space over the finite field GF(2). However, it is not possible to provide a similar representation for the Fano matroid using the real numbers in place of GF(2). A matrix A with entries in a field gives rise to a matroid M on its set of columns. The dependent sets of columns in the matroid are those that are linearly dependent as vectors. : This matroid is called the column matroid of A, and A is said to represent M. For instance, the Fano matroid can be represented in this way as a 3 × 7 (0,1) matrix. Column matroids are just vector matroids under another name, but there are often reasons to favor the matrix representation. A matroid that is equivalent to a vector matroid, although it may be presented differently, is called representable or linear. If M is equivalent to a vector matroid over a field F, then we say M is representable over in particular, M is real representable if it is representable over the real numbers. For instance, although a graphic matroid (see below) is presented in terms of a graph, it is also representable by vectors over any field. A basic problem in matroid theory is to characterize the matroids that may be represented over a given field F; Rota's conjecture describes a possible characterization for every finite field. The main results so far are characterizations of binary matroids (those representable over GF(2)) due to Tutte (1950s), of ternary matroids (representable over the 3 element field) due to Reid and Bixby, and separately to Seymour (1970s), and of quaternary matroids (representable over the 4 element field) due to . A proof of Rota's conjecture was announced, but not published, in 2014 by Geelen, Gerards, and Whittle. A regular matroid is a matroid that is representable over all possible fields. The Vámos matroid is the simplest example of a matroid that is not representable over any field. Matroids from graph theory A second original source for the theory of matroids is graph theory. Every finite graph (or multigraph) G gives rise to a matroid M(G) as follows: take as E the set of all edges in G and consider a set of edges independent if and only if it is a forest; that is, if it does not contain a simple cycle. Then M(G) is called a cycle matroid. Matroids derived in this way are graphic matroids. Not every matroid is graphic, but all matroids on three elements are graphic. • In a bipartite graph G = (U,V,E), one may form a matroid in which the elements are vertices on one side U of the bipartition, and the independent subsets are sets of endpoints of matchings of the graph. This is called a transversal matroid, and it is a special case of a gammoid. The transversal matroids are the dual matroids to the strict gammoids. A matroid that is equivalent to a matroid of this kind is called an algebraic matroid. The problem of characterizing algebraic matroids is extremely difficult; little is known about it. The Vámos matroid provides an example of a matroid that is not algebraic. == Basic constructions ==

There are some standard ways to make new matroids out of old ones. Duality If M is a finite matroid, we can define the orthogonal or dual matroid M^* by taking the same underlying set and calling a set a basis in M^* if and only if its complement is a basis in M. It is not difficult to verify that M^* is a matroid and that the dual of M^* is M. The dual can be described equally well in terms of other ways to define a matroid. For instance: • A set is independent in M^* if and only if its complement spans M. • A set is a circuit of M^* if and only if its complement is a coatom in M. • The rank function of the dual is r^*(S) = |S| - r(M) + r\left(E\smallsetminus S\right). According to a matroid version of Kuratowski's theorem, the dual of a graphic matroid M is a graphic matroid if and only if M is the matroid of a planar graph. In this case, the dual of M is the matroid of the dual graph of G. The dual of a vector matroid representable over a particular field F is also representable over F. The dual of a transversal matroid is a strict gammoid and vice versa. ;Example: The cycle matroid of a graph is the dual matroid of its bond matroid. Minors If M is a matroid with element set E, and S is a subset of E, the restriction of M to S, written M |S, is the matroid on the set S whose independent sets are the independent sets of M that are contained in S. Its circuits are the circuits of M that are contained in S and its rank function is that of M restricted to subsets of S. In linear algebra, this corresponds to restricting to the subspace generated by the vectors in S. Equivalently if T = M−S this may be termed the deletion of T, written M\T or M−T. The submatroids of M are precisely the results of a sequence of deletions: the order is irrelevant. The dual operation of restriction is contraction. If T is a subset of E, the contraction of M by T, written M/T, is the matroid on the underlying set E − T whose rank function is r'(A) = r(A \cup T) - r(T). In linear algebra, this corresponds to looking at the quotient space by the linear space generated by the vectors in T, together with the images of the vectors in E − T. A matroid N that is obtained from M by a sequence of restriction and contraction operations is called a minor of M. We say M contains N as a minor. Many important families of matroids may be characterized by the minor-minimal matroids that do not belong to the family; these are called forbidden or excluded minors. Sums and unions Let M be a matroid with an underlying set of elements E, and let N be another matroid on an underlying set F. The direct sum of matroids M and N is the matroid whose underlying set is the disjoint union of E and F, and whose independent sets are the disjoint unions of an independent set of M with an independent set of N. The union of M and N is the matroid whose underlying set is the union (not the disjoint union) of E and F, and whose independent sets are those subsets that are the union of an independent set in M and one in N. Usually the term "union" is applied when E = F, but that assumption is not essential. If E and F are disjoint, the union is the direct sum. == Additional terms == Let M be a matroid with an underlying set of elements E. • E may be called the ground set of M. Its elements may be called the points of M. • A subset of E spans M if its closure is E. A set is said to span a closed set K if its closure is K. • The girth of a matroid is the size of its smallest circuit or dependent set. • An element that forms a single-element circuit of M is called a loop. Equivalently, an element is a loop if it belongs to no basis. • An element that belongs to no circuit is called a coloop or isthmus. Equivalently, an element is a coloop if it belongs to every basis. : Loop and coloops are mutually dual. A simple matroid obtained from another matroid M by deleting all loops and deleting one element from each 2-element circuit until no 2 element circuits remain is called a simplification of M. A matroid is co-simple if its dual matroid is simple. • A union of circuits is sometimes called a cycle of M. A cycle is therefore the complement of a flat of the dual matroid. (This usage conflicts with the common meaning of "cycle" in graph theory.) • A separator of M is a subset S of E such that r(S) + r(E-S) = r(M). A proper or non-trivial separator is a separator that is neither E nor the empty set. An irreducible separator is a non-empty separator that contains no other non-empty separator. The irreducible separators partition the ground set E. • A matroid that cannot be written as the direct sum of two nonempty matroids, or equivalently that has no proper separators, is called connected or irreducible. A matroid is connected if and only if its dual is connected. • A maximal irreducible submatroid of M is called a component of M. A component is the restriction of M to an irreducible separator, and contrariwise, the restriction of M to an irreducible separator is a component. A separator is a union of components. • A matroid is called a paving matroid if all of its circuits have size at least equal to its rank. • The basis polytope P_M is the convex hull of the indicator vectors of the bases of M • The independence polytope of M is the convex hull of the indicator vectors of the independent sets of M. ==Algorithms==

Several important combinatorial optimization problems can be solved efficiently on every matroid. In particular: • Finding a maximum-weight independent set in a weighted matroid can be solved by a greedy algorithm. This fact may even be used to characterize matroids: if a family F of sets, closed under taking subsets, has the property that, no matter how the sets are weighted, the greedy algorithm finds a maximum-weight set in the family, then F must be the family of independent sets of a matroid. • The matroid partitioning problem is to partition the elements of a matroid into as few independent sets as possible, and the matroid packing problem is to find as many disjoint spanning sets as possible. Both can be solved in polynomial time, and can be generalized to the problem of computing the rank or finding an independent set in a matroid sum. • A matroid intersection of two or more matroids on the same ground set is the family of sets that are simultaneously independent in each of the matroids. The problem of finding the largest set, or the maximum weighted set, in the intersection of two matroids can be found in polynomial time, and provides a solution to many other important combinatorial optimization problems. For instance, maximum matching in bipartite graphs can be expressed as a problem of intersecting two partition matroids. However, finding the largest set in an intersection of three or more matroids is NP-complete. == Matroid software ==

Two standalone systems for calculations with matroids are Kingan's Oid and Hlineny's Macek. Both of them are open-sourced packages. "Oid" is an interactive, extensible software system for experimenting with matroids. "Macek" is a specialized software system with tools and routines for reasonably efficient combinatorial computations with representable matroids. Both open source mathematics software systems SAGE and Macaulay2 contain matroid packages. Maple has a package for dealing with matroids since the version 2024. ==Polynomial invariants==

There are two especially significant polynomials associated to a finite matroid M on the ground set E. Each is a matroid invariant, which means that isomorphic matroids have the same polynomial. Characteristic polynomial The characteristic polynomial of M – sometimes called the chromatic polynomial, • When M is the cycle matroid M(G) of a graph G, the characteristic polynomial is a slight transformation of the chromatic polynomial, which is given by χG (λ) = λcpM(G) (λ), where c is the number of connected components of G. • When M is the bond matroid M*(G) of a graph G, the characteristic polynomial equals the flow polynomial of G. • When M is the matroid M(A) of an arrangement A of linear hyperplanes in \mathbb{R}^n (or Fn where F is any field), the characteristic polynomial of the arrangement is given by pA (λ) = λn−r(M)pM (λ). Beta invariant The beta invariant of a matroid, introduced by Crapo (1967), may be expressed in terms of the characteristic polynomial p as an evaluation of the derivative : \beta(M) = (-1)^{r(M)-1} p_M'(1) or directly as : \beta(M) = (-1)^{r(M)} \sum_{X \subseteq E} (-1)^ r(X). The beta invariant is non-negative, and is zero if and only if M is disconnected, or empty, or a loop. Otherwise it depends only on the lattice of flats of M. If M has no loops and coloops then \beta( M ) = \beta( M^* ). : R_M(u,v) = \sum_{S\subseteq E} u^{r(M)-r(S)}v^{|S| - r(S)}. From this definition it is easy to see that the characteristic polynomial is, up to a simple factor, an evaluation of T_M, specifically, : p_M(\lambda) = (-1)^{r(M)} T_M(1-\lambda,0). Another definition is in terms of internal and external activities and a sum over bases, reflecting the fact that T(1,1) is the number of bases. This, which sums over fewer subsets but has more complicated terms, was Tutte's original definition. There is a further definition in terms of recursion by deletion and contraction. The deletion-contraction identity is : F(M) = F( M - e ) + F( M / e ) when e is neither a loop nor a coloop. An invariant of matroids (i.e., a function that takes the same value on isomorphic matroids) satisfying this recursion and the multiplicative condition : F(M \oplus M') = F(M) F(M') is said to be a Tutte–Grothendieck invariant. The Tutte polynomial T_G of a graph is the Tutte polynomial T_{ M(G) } of its cycle matroid. == Infinite matroids ==

The theory of infinite matroids is much more complicated than that of finite matroids and forms a subject of its own. For a long time, one of the difficulties has been that there were many reasonable and useful definitions, none of which appeared to capture all the important aspects of finite matroid theory. For instance, it seemed to be hard to have bases, circuits, and duality together in one notion of infinite matroids. The simplest definition of an infinite matroid is to require finite rank; that is, the rank of E is finite. This theory is similar to that of finite matroids except for the failure of duality due to the fact that the dual of an infinite matroid of finite rank does not have finite rank. Finite-rank matroids include any subsets of finite-dimensional vector spaces and of field extensions of finite transcendence degree. The next simplest infinite generalization is finitary matroids, also known as pregeometries. A matroid with possibly infinite ground set is finitary if it has the property that :x \in \operatorname{cl}(Y)\ \Leftrightarrow \ \text{ there is a finite set } Y' \subseteq Y \text{ such that } x \in \operatorname{cl}(Y'). Equivalently, every dependent set contains a finite dependent set. Examples are linear dependence of arbitrary subsets of infinite-dimensional vector spaces (but not infinite dependencies as in Hilbert and Banach spaces), and algebraic dependence in arbitrary subsets of field extensions of possibly infinite transcendence degree. Again, the class of finitary matroid is not self-dual, because the dual of a finitary matroid is not finitary. Finitary infinite matroids are studied in model theory, a branch of mathematical logic with strong ties to algebra. In the late 1960s matroid theorists asked for a more general notion that shares the different aspects of finite matroids and generalizes their duality. Many notions of infinite matroids were defined in response to this challenge, but the question remained open. One of the approaches examined by D.A. Higgs became known as B-matroids and was studied by Higgs, Oxley, and others in the 1960s and 1970s. According to a recent result by , it solves the problem: Arriving at the same notion independently, they provided five equivalent systems of axiom—in terms of independence, bases, circuits, closure and rank. The duality of B-matroids generalizes dualities that can be observed in infinite graphs. The independence axioms are as follows: • The empty set is independent. • Every subset of an independent set is independent. • For every nonmaximal (under set inclusion) independent set I and maximal independent set J, there is x \in J \smallsetminus I such that I \cup \{x\} is independent. • For every subset X of the base space, every independent subset I of X can be extended to a maximal independent subset of X. With these axioms, every matroid has a dual. ==History==

Matroid theory was introduced by . It was also independently discovered by Takeo Nakasawa, whose work was forgotten for many years (). In his seminal paper, Whitney provided two axioms for independence, and defined any structure adhering to these axioms to be "matroids". His key observation was that these axioms provide an abstraction of "independence" that is common to both graphs and matrices. Because of this, many of the terms used in matroid theory resemble the terms for their analogous concepts in linear algebra or graph theory. Almost immediately after Whitney first wrote about matroids, an important article was written by on the relation of matroids to projective geometry. A year later, noted similarities between algebraic and linear dependence in his classic textbook on Modern Algebra. In the 1940s Richard Rado developed further theory under the name "independence systems" with an eye towards transversal theory, where his name for the subject is still sometimes used. In the 1950s W.T. Tutte became the foremost figure in matroid theory, a position he retained for many years. His contributions were plentiful, including • the characterization of binary, regular, and graphic matroids by excluded minors • the regular-matroid representability theorem • the theory of chain groups and their matroids and the tools he used to prove many of his results: • the "Path theorem" • "Tutte homotopy theorem" (see, e.g., ) which are so complicated that later theorists have gone to great trouble to eliminate the need for them in proofs. and generalized to matroids Tutte's "dichromate", a graphic polynomial now known as the Tutte polynomial (named by Crapo). Their work has recently (especially in the 2000s) been followed by a flood of papers—though not as many as on the Tutte polynomial of a graph. In 1976 Dominic Welsh published the first comprehensive book on matroid theory. Paul Seymour's decomposition theorem for regular matroids () was the most significant and influential work of the late 1970s and the 1980s. Another fundamental contribution, by , showed why projective geometries and Dowling geometries play such an important role in matroid theory. By the 1980s there were many other important contributors, but one should not omit to mention Geoff Whittle's extension to ternary matroids of Tutte's characterization of binary matroids that are representable over the rationals , perhaps the biggest single contribution of the 1990s. In the current period (since around 2000) the Matroid Minors Project of Geelen, Gerards, Whittle, and others, has produced substantial advances in the structure theory of matroids. Many others have also contributed to that part of matroid theory, which (in the first and second decades of the 21st century) is flourishing. ==Researchers==

Mathematicians who pioneered the study of matroids include : Susumu Kuroda : Saunders MacLane : Richard Rado : Takeo Nakasawa : Hirokazu Nishimura : William T. Tutte : B. L. van der Waerden : Hassler Whitney Some of the other major contributors are : Jack Edmonds : Jim Geelen : Eugene Lawler : László Lovász : Gian-Carlo Rota : Paul D. Seymour : Dominic Welsh == Footnotes ==