with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel...
Transcript of with a focus on operations Congduan LiMatroid Theory with a focus on operations Congduan Li Drexel...
Matroid Theory
with a focus on operations
Congduan Li
Drexel University
November 21, 2011
Outline
MatroidMatroid ReviewDualContraction & DeletionExcluded Minors
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
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}
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}
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}
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}
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
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}
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}
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.
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
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 .
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
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
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
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*
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
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
Dual
Rank Function
� Rank function of the dual matroid M∗:
r∗(X ) = |X |+ r(E − X )− r(M).
Dual
Representable Case
� C =�Ir(M) Ar(M)×(|E |−r(M))
�
� C∗ =
�−AT I|E |−r(M)
�
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*
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.
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.
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
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 )
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.
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)}
Contraction & Deletion
Circuits
� Circuits:
C(M\T ) = {C ⊆ E − T : C ∈ C(M)}
C(M/T ) = min{C − T ⊆ E − T : C ∈ C(M)}
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.
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
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.
Excluded Minors
Summary
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.
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.