with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel...

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Matroid Theory with a focus on operations Congduan Li Drexel University November 21, 2011

Transcript of with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel...

Page 1: with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel University November 21, 2011 Outline Matroid Matroid Review Dual Contraction & Deletion

Matroid Theory

with a focus on operations

Congduan Li

Drexel University

November 21, 2011

Page 2: with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel University November 21, 2011 Outline Matroid Matroid Review Dual Contraction & Deletion

Outline

MatroidMatroid ReviewDualContraction & DeletionExcluded Minors

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Matroid Review

Definition

DefinitionA 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.

DemoAugmentation

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Matroid Review

Base/Basis

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

Properties

� Same cardinality for all bases;� The matroid is in the span of

any base;

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}

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Matroid Review

Base/Basis

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.

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}

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Matroid Review

Circuit

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 graph1

54

2

3

6

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

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Matroid Review

Circuit

A collection of sets C is the collectionof circuits of a matroid if and only if:

� φ /∈ C� if C1,C2 ∈ C with C1 ⊆ C2 then

C1 = C2

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

Consider the graph1

54

2

3

6

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

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Matroid Review

Rank Function

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 )

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

Page 9: with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel University November 21, 2011 Outline Matroid Matroid Review Dual Contraction & Deletion

Matroid Review

Closure

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}

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

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Matroid Review

Closure

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 and

y ∈ cl(X ∪ x)− cl(X ) thenx ∈ cl(X ∪ y)

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}

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Matroid Review

Greedy Algorithm

Ground set E , I is a collection ofsubsets of E . A mapping functionω : E → R. For X ∈ I

W (X ) =�

x∈X

ω(x)

An optimization solution in I is agreedy process:

1. X0 = 0, j = 02. If ∃ej+1 ∈ (E − Xj) such that

Xj ∪ ej+1 ∈ I and ej+1 has themax (min) weight among theelements left, thenXj+1 = Xj ∪ ej+1, j = j + 1,repeat. Otherwise, stop.

Such a collection I is thecollection of independent setsiff

1. φ ∈ I2. If I1 ∈ I and I2 ⊂ I1, then

I2 ∈ I3. For all weight mapping

functions, the abovegreedy algorithm is able tofind an optimizationsolution in I.

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Representable Matroids

Binary

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

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

Binary matroids have excluded minor: U2,4

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Representable Matroids

Ternary

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

Regular Matroids

� Representable over every field� Both Binary and Ternary

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 .

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Representable Matroids

Graphic Matroid

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 graphic matroids:U2,4,F7,F ∗

7 ,M∗(K5),M∗(K3,3)

K5 matroid and K3,3 matroid:

Figure: K5 matroid

Figure: K3,3 matroid

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Representable Matroids

Graphic

Graphic Matroids:� Independent sets: no cycles are contained� Rank function: r(X ) = |V (G [X ])|− w(G [X ])

� Flat: contains no cycle that is contained in but exactly oneedge

� Hyperplane: E (G )− H is a minimal set of edges whoseremoval from G increases the number of connectedcomponents

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Representable Matroids

Cographic and Planar

A matroid is called cographic if itsdual is graphic.Ex: M

∗(K5) and M∗(K3,3)

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.

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

7 ,M(K5),M(K3,3)Excluded minors for planar:those excluded minors ofgraphic and cographic

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Dual

Definition

The dual matroid M∗ to a matroid M is the matroid with bases

that are complements of the bases of M.

B(M∗) = {E − B|B ∈ B(M)}.

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

1

5

4

632

45

61

23

G G*

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Dual

Independent Sets

� X is independent if and only if E − X is cospanning� X is spanning if and only if E − X is coindependent

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Dual

Circuits

� X is a circuit if and only if E − X is a cohyperplane� X is a hyperplane if and only if E − X is a cocircuit

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Dual

Rank Function

� Rank function of the dual matroid M∗:

r∗(X ) = |X |+ r(E − X )− r(M).

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Dual

Representable Case

� C =�Ir(M) Ar(M)×(|E |−r(M))

� C∗ =

�−AT I|E |−r(M)

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Dual

Graphic Case

� Circuits X of M∗(G ) are cocircuits of M(G ); E − X is a

hyperplane in M(G)� Complement of a Hyperplane is a minimal collection of edges

(Bonds) whose removal from G increases the number ofconnected components

� Circuits of M∗(G ) are the bonds of G

1

5

4

632

45

61

23

G G*

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Contraction & Deletion

Definition

� Restriction is a shrink of domain; Deletion a subset T is arestriction on (E-T).

M\T = M|(E − T )

I(M\T ) = {I ⊆ T : I ∈ I(M)}C(M\T ) = {C ⊆ T : C ∈ C(M)}rM\T (X ) = rM(X )Example.

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Contraction & Deletion

Definition & Graphic case

� Contraction is dual operation of deletion.

M/T = (M∗\T )∗

In a graphic matroid, M(G )/T = M(G/T ), which means thematroid contracting T is the matroid on the subgraph obtainedfrom G by contracting edges T .Example.

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Contraction & Deletion

Representable case

Intuitively, contraction means delete elements (edges) and mergethe corresponding end points into single points if viewed in a graph.While deletion of an element does not do the combination.In Vectors, contracting an element means remove thecorresponding component from the other vectors.Contracting T from a representable matroid: Find a base B ⊇ T

and represent other vectors with B, then delete T .

V1

V2V3

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Contraction & Deletion

Rank Function

� Rank function:rM\T (X ) = rM(X )

rM/T (X ) = rM(X ∪ T )− rM(T )

Also compare these with dual operation, since the above equationcan be obtained from rM\T and the following one

r∗(X ) = |X |+ r(E − X )− r(E )

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Contraction & Deletion

Independent sets

� Independent sets

I(M\T ) = {I ⊆ E − T : I ∈ I(M)}

I(M/T ) = {I ⊆ E−T : I∪BT ∈ I(M)} = {I ∈ I(M) : I∩clM(X ) = φ}

where BT is the base of M|T . M|T has a base B such thatB ∪ I ∈ I.

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Contraction & Deletion

Bases

� Bases: B(M\T ) is the set of maximal members of{B − T : B ∈ B(M)};

B(M/T ) = {B � ⊆ E − T : B� ∪ BT ∈ B(M)}

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Contraction & Deletion

Circuits

� Circuits:

C(M\T ) = {C ⊆ E − T : C ∈ C(M)}

C(M/T ) = min{C − T ⊆ E − T : C ∈ C(M)}

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Contraction & Deletion

Closure

� Closure:clM\T (X ) = clM(X )− T

clM/T (X ) = clM(X ∪ T )− T

The closures also show the difference between deletion andcontraction. That is, deletion does not affect the relationshipsbetween remaining elements.

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Contraction & Deletion

Other theorems

� M\T = M/T iff r(T ) + r(E − T ) = r(M), T is a separatorof M.

� M\e = M/e iff e is a loop or a coloop of M

� In a matroid M, let T1 and T2 be disjoint subsets of E (M).Then TFAE:

� (M\T1)\T2 = M\(T1 ∪ T2) = (M\T2)\T1� (M/T1)/T2 = M/(T1 ∪ T2) = (M/T2)/T1� (M\T1)/T2 = (M/T2)\T1

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Excluded Minors

Definition

DefinitionMinor: If a matroid M

� can be obtained by operating combinationof restrictions and contractions on a matroid M, then M

� is called aminor of M.Examples: U2,4 is excluded by binary; F7&F

∗7 are excluded by

ternary.

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Excluded Minors

Summary

Relationships between various classes of matroids:

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>-3/+$&-!?#-&!B!@&-A?#-&!

2%#-):+%)#/!

@&-A9#,,4)!

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.

Regular matroid can be represented by every field.A binary matroid is regular if it can be represented by some otherfield other than that of characteristic 2.