Matroid

A matroid ( n ) is a mathematical structure with which the concept of independence is generalized from linear algebra. Matroids have applications in many areas of combinatorics, in particular of the combinatorial optimization and graph theory.

Definition

Be a lot and a lot of system. The pair is called matroid if the following conditions are met:

Here is the cardinality of the set.

The elements of the hot points, the elements of the independent subsets of, quantities of are called dependent.

Satisfies only the first two properties, thus making it an independence system.

The smallest that satisfies the fourth condition is called the rank of the matroid.

Exchange property

The unique selling point of a matroid over an ordinary independence system consists in the replacement property, so the third condition above definition. During the first two axioms can be understood as a claim of the existence of at least one independent quantity and the seclusion of the system with respect to the inclusion of light, so the motivation of the exchange property is less obvious.

One can illustrate this as follows to: By applying the exchange property located points can be an independent set to a (smaller) independent set to add. That is why it is also called the magnification property (of English. Augmentation property ). From the amount thus generated, we know that it is also independent again. Although now it contains elements of the set, but the other points are replaced by elements contained in comparison to the latter. This in turn justifies the name exchange property.

Examples

• The prototype of a matroid form the columns of a matrix - hence the name - or any other finite vector system with the totality of linearly independent subsystems as independent subsets.
• The edge set of a finite graph together with the circle-free subsets form the so-called graphic matroid. Its maximal independent sets are then exactly the spanning forests of the underlying graph.

Alternative characterizations of a matroid

The applications of matroids in the various mathematical disciplines have motivated the definition of other set systems over or functions on the base amount. Conversely characterize these the underlying matroid complete, ie from each of the new systems can be back a lot so define that becomes a matroid.

Circles

Be a quantity system, hot loop system if it satisfies the following conditions:

The minimal dependent subsets of a matroid always be a closed system. This is particularly clear in graphic matroids, since there are elements of precisely the cycles of the underlying graphene, where the term originated.

A lot is exactly independent if it is acyclic. Thus, if a non-empty circuit system over a set, so is the other way around with a matroid such that the minimal elements of just the circles are.

Bases

Be back a lot of system, it is called if the following holds basic system:

The maximal independent subsets of a matroid form a base system. For finite dimensional vector spaces, this coincides with the classical base term and the defining property corresponds to the Austauschlemma of Steinitz.

It can be shown that all elements of the same thickness must possess, it is equal to the rank of the matroid defined above.

In turn, is a non-empty base system over a lot, so this is a matroid and the maximal elements in are exactly the bases of.

Rank function

An illustration is a rank function if it satisfies the following properties:

Alternatively, one finds the following equivalent definition:

Is back for a matroid and a subset of a (sub - ) matroid. It was his rank (alternatively ), the ranking function is. In the first example above corresponds to the rank of a set of vectors of the dimension of its linear span.

Analogously, the subsets with a ranking function in the above sense a system of independent quantities and is used to rank functions of the resulting matroid.

Closure operator

For a matroid with the rank function is due to a subset of a closure operator with the additional property

Explained. It corresponds to the formation of the linear hull in the example above.

Is a closure operator over the crowd with this custom property, the basic amount with independent subsets as a system becomes a matroids whose associated closure operator is again.

Greedy algorithms

A weighted matroid is a matroid with weight function. For these matroids calculate greedy algorithms always bases with minimum or maximum weight. An example is the algorithm of Kruskal for computing a minimum spanning forest of an edge- weighted graph.

A Unabhängigskeitssystem is reversed if and only a matroid if a greedy algorithm to each weight function can always calculate bases with minimum / maximum weight.