Introduction to Matroid Theory - Faculty & Staff€¦ · cycle-free graph is cycle-free; 3...

Fundamentals of Matroid

Some Classes of Representable Matroids

Summary

Introduction to Matroid Theory

Congduan Li

Adaptive Signal Processing and Information Theory Research Group

ECE Department, Drexel University

November 21, 2011

Congduan Li Introduction to Matroid Theory



Summary

Outline

1 Fundamentals of MatroidGeneral Definition of MatroidsEquivalent DefinitionsOperations

2 Some Classes of Representable MatroidsRepresentable MatroidsExcluded Minors SummaryRelationships between Various Classes of Matroids

3 Summary




Summary

Outline

1 Fundamentals of Matroid

General Definition of Matroids

Equivalent Definitions

Operations

2 Some Classes of Representable Matroids

Representable Matroids

Excluded Minors Summary

Relationships between Various Classes of Matroids

3 Summary




Summary



Operations

General Definition of MatroidsVector Independence

Properties of Vector Independence

1 φ is independent to any vector;

2 Hereditary: subset of someindependent vectors are alsoindependent;

3 Augmentation: we can add anew vector to a smallerindependent set to keepindependency.

Example

Let A be the following matrixover field R of real numbers.

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

Hereditary: {1, 2, 3} → {1, 2}Augmentation:{1, 2}, {2, 3, 5} → {1, 2, 3}




Summary



Operations

General Definition of MatroidsCycles in Graphs

Properties of Graphs

1 φ contains no cycle;

2 Hereditary: subgraph of acycle-free graph is cycle-free;

3 Augmentation: Add a new edgeto a smaller cycle-free subgraph.

Example

Graph associated with A1

54

2

3

6

Example

Let A, B be the followingsubgraphs.

4

13

4

12

6

5

7

Hereditary:{1, 3, 4, 5} → {1, 3, 4}Augmentation:{1, 4}, {1, 3, 4, 5} → {1, 3, 4}




Summary



Operations

General Definition of MatroidsCollinear & Coplanar in Geometry

Geometry Diagram

1 Collinear/planar points aredependent (at least 3);

2 φ is not collinear/coplanar toany point;

3 Hereditary: subset of somenon-collinear/coplanar pointsare still non-collinear/coplanar;

4 Augmentation: we can add anew point to a smaller set ofsome non-collinear/coplanarpoints.

Example

See the following geometrydiagram.

1

5

2 43

6

Associated with

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

Generalize=⇒Matroid




Summary



Operations

General Definition of MatroidsIndependent sets

Definition

A matroid M is an ordered pair (E , I)consisting of a finite set E and acollection I of subsets of E having thefollowing three properties:

1 φ ∈ I;2 Hereditary:

If I ∈ I and I � ⊆ I , then I � ∈ I;3 Augmentation:

If I1 and I2 are in I and |I1| <|I2|, then there is an element e of I2−I1 such that I1 ∪ e ∈ I.

Demo

I is a collection of subset of E

Hereditary




Summary



Operations

General Definition of MatroidsIndependent sets

Definition

A matroid M is an ordered pair (E , I)consisting of a finite set E and acollection I of subsets of E having thefollowing three properties:

1 φ ∈ I;2 Hereditary:

If I ∈ I and I � ⊆ I , then I � ∈ I;3 Augmentation:

If I1 and I2 are in I and |I1| <|I2|, then there is an element e of I2−I1 such that I1 ∪ e ∈ I.

Demo

Augmentation




Summary



Operations

Outline




Operations





3 Summary




Summary



Operations

Equivalent DefinationsBase/Basis

Definition

Bases B(M) of a matroid are themaximal independent sets.

Properties

Same cardinality for all bases;

The matroid is a span of anybase;

Example

Consider

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

Bases: {1,2,3}, {1,2,4},{1,2,5}, {1,3,4}, {1,4,5},{2,3,4}, {2,3,5}, {3,4,5}




Summary



Operations

Equivalent DefinationsBase/Basis

Definition

A collection of subsets B ⊆ 2E of aground set E are the bases of amatroid if and only if:

B is non-empty;

base exchange: If B1,B2 ∈ B,for any x ∈ (B1 − B2), there isan element y ∈ (B2 − B1) suchthat (B1 − x) ∪ y ∈ B.

Example

Consider

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

Bases: {1,2,3}, {1,2,4},{1,2,5}, {1,3,4}, {1,4,5},{2,3,4}, {2,3,5}, {3,4,5}




Summary



Operations

Equivalent DefinitionsCircuit

Definition

The circuits C(M) of a matroid arethe minimal dependent sets.

Property

If any one element is deletedfrom a circuit, it will become anindependent set.

Independent sets I(M) does notcontain any element in C(M).

Consider the graph

1

54

2

3

6

Circuits: {1,2,3,4}, {1,3,5},{2,4,5}, {6}




Summary



Operations

Equivalent DefinitionsCircuit

Definition

A collection of sets C is thecollection of circuits of a matroid ifand only if:

φ /∈ Cif C1,C2 ∈ C with C1 ⊆ C2 thenC1 = C2

if C1,C2 ∈ C,C1 �= C2 ande ∈ C1 ∩ C2 then there is someC3 ∈ C with C3 ⊆ (C1∪C2)− e.

Consider the graph

1

54

2

3

6

Circuits: {1,2,3,4}, {1,3,5},{2,4,5}, {6}




Summary



Operations

Equivalent DefinitionsRank Function

Definitions

Rank: maximun number of linearlyindependent vectors.For any base of a matroid,r(B) = r(M).Formally, a rank function isr : 2E → N ∪ {0}, for which

For X ⊆ E , 0 ≤ r(X ) ≤ |X |If X ⊆ Y ⊆ E , r(X ) ≤ r(Y )

∀X ,Y ⊆ E , r(X ∩ Y ) + r(X ∪Y ) ≤ r(X ) + r(Y )

Example

Consider

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

Bases: {1,2,3}, {1,2,4},{1,2,5}, {1,3,4}, {1,4,5},{2,3,4}, {2,3,5}, {3,4,5};Rank: r(M) = r(B) = 3




Summary



Operations

Equivalent DefinitionsClosure

Definitions

Given a rank function of a matroid,the closure operation cl : 2E → 2E isdefined as

cl X = {x ∈ E |r(X ∪ x) = r(X )}.

Closure is a span of a subspace.cl B = E (M);Independent Set:I = {X ⊆ E |x /∈ cl(X − x)∀x ∈ X}

Definition

A subset X of E (M) is said tobe a flat if cl X = X.

Example

Consider

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

cl {1,2}={1,2,} → flatcl {1,3}={1,3,5}




Summary



Operations

Equivalent DefinitionsClosure

Definitions

A function cl : 2E → 2E is closure fora matroid M = (E , I) iff

If X ⊆ E then X ⊆ clX

If X ⊆ Y ⊆ E , then clX ⊆ clY

If X ⊆ E then cl(cl(X )) = clX

If X ⊆ E and x ∈ E andy ∈ cl(X ∪ x)− cl(X ) thenx ∈ cl(X ∪ y)

Example

Consider

A =

1 2 3 4 5 6

1 0 0 1 10 0 1 1 10 1 0 1 0

000

cl {1,2}={1,2,} → flatcl {1,3}={1,3,5}




Summary



Operations

Outline




Operations





3 Summary




Summary



Operations

OperationsDual

Definition

The dual matroid M∗ to a matroidM is the matroid with bases that arecomplements of the bases of M.

B(M∗) = {E − B|B ∈ B(M)}.

(M∗)∗ = M and X is a base ifand only if E − X is a cobase.

Rank function of the dualmatroid M∗:

r∗(X ) = |X |+ r(E − X )− r(M).

Example

1

5

4

632

45

61

23

G G*




Summary



Operations

OperationsContraction, Restriction, Minor

Definitions

Restriction of M to X (X ⊂ E ) isthe matroid M � with ground set Xand independent setsI � = {A ∈ I|A ⊂ X}. We denote itas M|X .

Definitions

Deletion: M\T is therestriction of M on E − T ;M\T = M|(E − T )

Example




Summary



Operations

OperationsContraction, Restriction, Minor

Definitions

Contraction is the dual operation ofdeletion. M/T = (M∗\T )∗

Definition

Minor: If a matroid M � can beobtained by operatingcombination of restrictions andcontractions on a matroid M,then M � is called a minor of M.




Summary



Operations

OperationsExcluded Minor

Definition

Minor-closed matroids are a class ofmatroids of which every minor of amember is also in the class.

Minor-closed classes of matroidshave excluded minorcharacterization, that is,matroids that are not in theclass but have all their properminors in the class.

Example

If a matroid contains U2,4 as aminor, it cannot berepresented by binary vectorspace. U∗

2,4 = U2,4.




Summary



Operations

Outline




Operations





3 Summary




Summary




Representable MatroidsBinary

Definition

A matroid that is isomorphic to avector matroid, where all elementstake values from field F, is calledF-representable.

Example

Fano matroid is onlyrepresentable over field ofcharacteristic 2 (Binary).

Binary matroids have excluded minor: U2,4




Summary




Representable MatroidsTernary

Definition

Ternary matroids are those who canbe represented over field withcharacteristic 3.

Regular Matroids

Representable over every field

Both Binary and Ternary

Example

U2,k is F-representable if andonly if |F| ≥ k − 1. |U2,4| is3-representable (Ternary).

B =1 2 3 4�1 0 1 −10 1 1 1

�

Ternary matroids have excluded minor: U2,5,U3,5,F7 and F ∗7.




Summary




Representable MatroidsGraphic Matroid

Definition

In a graph G = (V ,E ), let C ⊆ 2E

be the sets of the graph cycles. ThenC forms a set of circuits for a matroidwith ground set E . The matroidM(G ) derived in this manner is calleda cycle matroid (Graphic Matroid).

Excluded minors for graphicmatroids:U2,4,F7,F ∗

7,M∗(K5),M∗(K3,3)

Example

K5 matroid and K3,3 matroid:

Figure: K5 matroid

Figure: K3,3 matroid




Summary




Representable MatroidsCographic and Planar

Definition

A matroid is called cographic if itsdual is graphic.Ex: M∗(K5) and M∗(K3,3)

Definitions

A both graphic and cographicmatroid is called planar, isomorphicto a cycle matroid derived from aplanar. In a planar, all edges can bedrawn on a plane withoutintersections.

Example

Excluded minors for cographic:U2,4,F7,F ∗

7,M(K5),M(K3,3)

Excluded minors for planar:those excluded minors ofgraphic and cographic




Summary




Outline




Operations





3 Summary




Summary




Excluded MinorsVarious Classes of Representable Matroids

Excluded Minors:

Matroid Class Excluded Minor(s)Binary U2,4

Ternary U2,5,U3,5,F7,F ∗7

Regular U2,4,F7,F ∗7

Graphic U2,4,F7,F ∗7,K ∗

5,K ∗

3,3

Cographic U2,4,F7,F ∗7,K5,K3,3

Planar U2,4,F7,F ∗7,K5,K3,3,K ∗

5,K ∗

3,3




Summary




Outline




Operations





3 Summary




Summary




Relationships

Relationships between various classes of matroids:

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Figure: Relationships between various classes matroids: U2,4,F7, etc. inthe graph are examples of the matroid class in which they lie. If anexample is not contained in one class, it is called an excluded minor ofthis class.




Summary

Summary

Today’s talk

1 Some Equivalent Definitions of Matroids: Sets, Base, Circuit,Rank, Closure

2 Operations of Matroids: Dual, Restriction, Contraction, Minor

3 Representable Matroids: Binary, Ternary, Regular, Graphic,Cographic, Planar

4 Excluded Minors and relationships between various classes ofmatroids




Summary

Q&A

Thank you!

Next: Matroid, Network and Coding.

James Oxley, Matroid Theory, Oxford University Press, 2011


Introduction to Matroid Theory - Faculty & Staff€¦ · cycle-free graph is cycle-free; 3...

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