A matroid is a combinatorial structure whose properties imitate those of linearly independent subsets of a vector space. Notions such as rank and independence (of a subset) have a meaning for any matroid, as does the notion of duality.
1 Definitions of a matroid
First rank definition
A matroid consists of a set and a function satisfying the axioms:
In this situation, is called the rank function of the matroid . If every singleton of has rank equal to 1, then is called a normal matroid.
Second rank definition
A matroid consists of a set and a function satisfying the axioms:
If and , then is either 0 or 1.
If , and , then implies .
Independent set definition
A matroid is a pair with (called the independent sets of ) satisfying the axioms:
The matroid is normal if every singleton in is independent.
A matroid is a pair with (called the bases of ) a subset of satisfying the axioms:
has at least one base.
The proper subsets of a base are not bases.
If and are bases and , then for some , the set is a base.
The matroid is called normal if each singleton of is contained in a base of .
A matroid consists of a set and a function , called the closure operator, satisfying the axioms:
Any subset of is contained in its closure.
If then .
If is in the closure of but not that of , then is in the closure of .
The closure operator is sometimes also called the span mapping of the matroid. In this case is called the span of .
The matroid is normal if the empty set is its own closure.
A matroid is a pair with (called the circuits of ) satisfying the axioms:
The matroid is normal if no singleton of is a circuit.
Combinatorial optimization definition
There’s yet another definition of matroids from “Combinatorial Optimization” by Papadimitriou and Steiglitz. pp. 280-285.
It requires three definitions to generate their definition of matroids. These are more or less grabbed from the book above.
Definition 2: The “combinatorial optimization problem” (nonstandard term used in book) is as follows. Let (E,g) be a subset system and weight w, a nonnegative real function on E. Find the subset in g with the largest total weight.
Definition 3: Let (E,g) be a subset system and w, a weight function defined as above to give the “combinatorial optimization problem”. The greedy algorithm for construction of a subset I in g is as follows. Start with I being the empty set. Take the next highest weight element, e in E (w(e) ¿= w(f) for all f in E). If the union of I and e is in g, then add element e to I. Repeat until you exhaust all elements of E.
Now we have the definition of a matroid.
Definition 4:Let M=(E,g) be a subset system. M is a “matroid” if the greedy algorithm correctly solves the ”combinatorial optimization problem” for any weight function associated with M.
2 Equivalence of the definitions
It would take several pages to spell out what is a circuit in terms of rank, and likewise for each other possible pair of the alternative defining notions, and then to prove that the various sets of axioms unambiguously define the same structure. So let me sketch just one example: the equivalence of Definitions 1 (on rank) and 6 (on circuits). Assume first the conditions in Definition 1. Define a circuit as a minimal subset of having the property . With a little effort, we verify the axioms (c1)-(c3). Now assume (c1)-(c3), and let be the largest integer such that has a subset for which
– contains no element of
One now proves (r1)-(r3). Next, one shows that if we define in terms of , and then another rank function in terms of , we end up with =. The equivalence of (r*) and (c*) is easy enough as well.
3 Examples of matroids
Let be a vector space over a field , and let be a finite subset of . For , let be the dimension of the subspace of generated by . Then is a matroid. Such a matroid, or one isomorphic to it, is said to be representable over . The matroid is normal iff . There exist matroids which are not representable over any field.
The second example of a matroid comes from graph theory. The following definition will be rather informal, partly because the terminology of graph theory is not very well standardised. For our present purpose, a graph consists of a finite set , whose elements are called vertices, plus a set of two-element subsets of , called edges. A circuit in the graph is a finite set of at least three edges which can be arranged in a cycle:
such that the vertices are distinct. With circuits thus defined, satisfies the axioms in Definition 6, and is thus a matroid, and in fact a normal matroid. (The definition is easily adjusted to permit graphs with loops, which define non-normal matroids.) Such a matroid, or one isomorphic to it, is called “graphic”.
Let be a finite set, where and are nonempty and disjoint. Let a subset of . We get a “matching” matroid on as follows. Each element of defines a “line” which is a subset (a row or column) of the set . Let us call the elements of “points”. For any let be the largest number such that for some set of points :
– No two points of are on the same line
– Any point of is on a line defined by an element of .
4 The dual of a matroid
Proposition: Let be a matroid and its rank function. Define a mapping by
Then the pair is a matroid (called the dual of .
We leave the proof as an exercise. Also, it is easy to verify that the dual of the dual is the original matroid. A circuit in is also referred to as a cocircuit in . There is a notion of cobasis also, and cospan.
If the dual of is graphic, is called cographic. This notion of duality agrees with the notion of same name in the theory of planar graphs (and likewise in linear algebra): given a plane graph, the dual of its matroid is the matroid of the dual graph. A matroid that is both graphic and cographic is called planar, and various criteria for planarity of a graph can be extended to matroids. The notion of orientability can also be extended from graphs to matroids.
5 Binary matroids
A matroid is said to be binary if it is representable over the field of two elements. There are several other (equivalent) characterisations of a binary matroid , such as:
– For any circuit and cocircuit , we have .
Any graphic matroid is binary. The dual of a binary matroid is binary.
The definition of the chromatic polynomial of a graph,
extends without change to any matroid. This polynomial has something to say about the decomposibility of matroids into simpler ones.
Also on the topic of decomposibility, matroids have a sort of structure theory, in terms of what are called minors and separators. That theory, due to Tutte, goes by induction; roughly speaking, it is an adaptation of the old algorithms for putting a matrix into a canonical form.
7 Further reading
A good textbook is:
James G. Oxley, Matroid Theory, Oxford University Press, New York etc., 1992
plus the updates-and-errata file at Dr. Oxley’s http://www.math.lsu.edu/ oxleywebsite.
The chromatic polynomial is not discussed in Oxley, but see e.g. http://www.math.binghamton.edu/zaslav/MatroidsZaslavski.
|Date of creation||2013-03-22 13:08:56|
|Last modified on||2013-03-22 13:08:56|
|Last modified by||mps (409)|