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Transversal matroids

Anna de Mier

Universitat Politecnica de Catalunya

Barcelona, Spain

Let’s Matroid CUSO doctoral school, Neuchatel, March 25–27, 2015

What’s ahead

I Definition, first properties, and examples

I Further properties of transversal matroids

I Characterizations of transversal matroids

I Other topics

Definition, first properties, and examples

Keywords: set systems, partial transversals, transversal matroids,matrix representations, lattice path matroids, bicircular matroids

Transversals of set systemsA set system is a collection A of subsets of a set S :

A = (Aj : j ∈ J)

(both S and J finite)

Ex S = [9], A = (A,B,C ,D) :

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

Set systems are in bijection with bipartite graphs with stable partsJ and S

A CB D

4 5 6 7 8 92 31

Transversals of set systemsA set system is a collection A of subsets of a set S :

A = (Aj : j ∈ J)

(both S and J finite)

Ex S = [9], A = (A,B,C ,D) :

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

Set systems are in bijection with bipartite graphs with stable partsJ and S

A CB D

4 5 6 7 8 92 31

Transversals of set systems

Def A partial transversal of A = (Aj : j ∈ J) is a subset T of Ssuch that there is an injective map ϕ : T → J such that

t ∈ Aϕ(t) for all t ∈ T

If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)

{6, 7, 8} is a partial transversal{5, 6, 7, 8} is a transversal

If |T | = |J|, partial transversals are called just transversals (akasystems of distinct representatives)Ex

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

{6, 7, 8} is a partial transversal

{5, 6, 7, 8} is a transversal

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

{6, 7, 8} is a partial transversal

{5, 6, 7, 8} is a transversal

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

In terms of the associated bipartite graph, partial transversals arethe end-vertices in S of partial matchings of the graph

A CB D

4 5 6 7 8 92 31

{2, 5, 8, 9} is a partial transversal that corresponds to severalmatchings

A CB D

4 5 6 7 8 92 31

{2, 5, 8, 9} is a partial transversal

that corresponds to severalmatchings

A CB D

4 5 6 7 8 92 31

{2, 5, 8, 9} is a partial transversal that corresponds to severalmatchings

Thm (Edmonds and Fulkerson 65)The partial transversals of a set system A are the independent setsof a matroid

This matroid is called a transversal matroid, denoted M[A]

The set system A is a presentation of the matroid

Proof: it will become transparent once we introduce another viewon set systems and transversals. But first let’s look at someexamples

Transversal matroids: examples

I All uniform matroids Ur ,n are transversal

Take A = ([n], [n], (r). . ., [n])

I But many other presentations are possible

U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})

I Which transversal matroid does the following set system give?

({1 2 4}, {2 3}, {4 5})

A rank-3 matroid with 2 lines with a common point, aka thecycle matroid of K4 minus an edge

I Transversal matroids are closed under deletion

Take A = ([n], [n], (r). . ., [n])

U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})

({1 2 4}, {2 3}, {4 5})

Take A = ([n], [n], (r). . ., [n])

U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})

({1 2 4}, {2 3}, {4 5})

Take A = ([n], [n], (r). . ., [n])

U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})

({1 2 4}, {2 3}, {4 5})

Take A = ([n], [n], (r). . ., [n])

U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})

({1 2 4}, {2 3}, {4 5})

Take A = ([n], [n], (r). . ., [n])

U3,6 = ({1 2 3 4}, {1 2 5 6}, {3 4 5 6})

({1 2 4}, {2 3}, {4 5})

The matrix perspective

Given a set system (Aj , j ∈ J) on S consider a |J| × |S | matrixwith entries

mi ,j =

{0 if j 6∈ Ai

xij if j ∈ Ai

where the xij are algebraically independent numbers

Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}x11 x12 0 0 0 x16 0 0 x190 x22 x23 x24 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49

Which square submatrices have non-zero determinant?

Given a set system (Aj , j ∈ J) on S consider a |J| × |S | matrixwith entries

mi ,j =

{0 if j 6∈ Ai

xij if j ∈ Ai

where the xij are algebraically independent numbers

Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}x11 x12 0 0 0 x16 0 0 x190 x22 x23 x24 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49

Which square submatrices have non-zero determinant?

Which square submatrices have non-zero determinant?∣∣∣∣∣∣∣∣0 0 0 x19

x25 x27 0 x29x35 0 x38 x390 x47 x48 x49

∣∣∣∣∣∣∣∣ =

− x25x47x38x19 − x35x27x48x19 6= 0

I For T ⊆ S and ϕ : T → J, observe

t ∈ Aϕ(t) for all t ∈ T⇔∏t∈T

xϕ(t),t 6= 0

I As all of the xij are algebraically independent,T is a partial transversal ⇔ the columns corresponding to T

are linearly independent

x25 x27 0 x29x35 0 x38 x390 x47 x48 x49

∣∣∣∣∣∣∣∣ = − x25x47x38x19 − x35x27x48x19 6= 0

xϕ(t),t 6= 0

x25 x27 0 x29x35 0 x38 x390 x47 x48 x49

∣∣∣∣∣∣∣∣ = − x25x47x38x19 − x35x27x48x19 6= 0

xϕ(t),t 6= 0

x25 x27 0 x29x35 0 x38 x390 x47 x48 x49

∣∣∣∣∣∣∣∣ = − x25x47x38x19 − x35x27x48x19 6= 0

xϕ(t),t 6= 0

Transversal matroids are representable

Thus, the partial transversals of a set systems are indeed theindependent sets of a matroid, and this matroid is representable

A stronger result is

Thm (Piff and Welsh 70)Every transversal matroid is representable over all sufficiently largefields

Transversal matroids are representable

Thus, the partial transversals of a set systems are indeed theindependent sets of a matroid, and this matroid is representable

A stronger result is

Thm (Piff and Welsh 70)Every transversal matroid is representable over all sufficiently largefields

The rank of a transversal matroid

Suppose s1, . . . , sr are a basis of M = M[A], and that si ∈ Ai

The matrix looks like(X Y

)where det(X ) 6= 0

Thus, the first r rows span the row-space, so M = M[(A1, . . . ,Ar )]

ThmA rank-r transversal matroid always has a presentation with r setsMorevoer, if there are no coloops, all presentations have exactly rsets

)where det(X ) 6= 0

ThmA rank-r transversal matroid always has a presentation with r sets

Morevoer, if there are no coloops, all presentations have exactly rsets

)where det(X ) 6= 0

Example: lattice path matroids

Lattice paths start at the origin and have steps E = (1, 0) andN = (0, 1)

We label each possible N step by its distance to the originConsider two non-crossing paths P and Q with common endpoints

We label each possible N step by its distance to the origin

Consider two non-crossing paths P and Q with common endpoints

We label each possible N step by its distance to the origin

Consider two non-crossing paths P and Q with common endpoints

Let P be the set of lattice paths in the region bounded by P and Q

N2 = {2 3 4 5 6}

Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level iEach path in P gives a transversal of (N1, . . . ,Nr )

N2 = {2 3 4 5 6}

Each such path is determined by the labels of its N steps

Let Ni be the set of possible N steps at level iEach path in P gives a transversal of (N1, . . . ,Nr )

N2 = {2 3 4 5 6}

Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level i

Each path in P gives a transversal of (N1, . . . ,Nr )

N4 = {7 8 9}N3 = {4 5 6 7}N2 = {2 3 4 5 6}N1 = {1 2 3}

Each such path is determined by the labels of its N stepsLet Ni be the set of possible N steps at level i

Each path in P gives a transversal of (N1, . . . ,Nr )

N4 = {7 8 9}N3 = {4 5 6 7}N2 = {2 3 4 5 6}N1 = {1 2 3}

Example: lattice path matroidsThm (Bonin, de Mier and Noy 2003)Let P and Q be two non-crossing lattice paths ending at (m, r)and consider the transversal matroid M[P,Q] with presentation(N1, . . . ,Nr )Then M[P,Q] has rank r , m + r elements and its bases are inbijection with lattice paths in P

Proof idea:

A lattice path matroid is a matroid isomorphic to M[P,Q] forsome P,Q

Proof idea:

Can different pairs of paths give isomorphic lattice path matroids?

Turning the diagram 180 degrees around gives isomorphic matroids

Thm (Bonin and de Mier 06)The case described above is the only situation where differentpaths yield isomorphic lattice path matroids

Example: bicircular matroids

Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}

M[(Av : v ∈ V )] is a transversal matroid on E

Who are its independent sets?

Edge-sets spanning trees or unicyclic graphs are independent:

Let G = (V ,E ) be a graphFor v ∈ V , let Av = {e ∈ E : e is incident with v}M[(Av : v ∈ V )] is a transversal matroid on E

��

Thm (Matthews 77, Simoes-Pereira 72)The independent sets of the matroid M[(Av : v ∈ V )] are theedge-sets all whose components are trees or unicyclic.

The circuits correspond to the bicycles of the graph

Note that bicircular matroids are loopless

Thm (Matthews 77, Simoes-Pereira 72)The independent sets of the matroid M[(Av : v ∈ V )] are theedge-sets all whose components are trees or unicyclic.The circuits correspond to the bicycles of the graph

Further properties of transversal matroids

Keywords: geometric representation, cyclic flats, contraction,duality, maximal and minimal presentations

Geometric representation of transversal matroids

Consider a matrix representation with non-negative entries andwhere all column sums are 1:

Ex A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}1 x12 0 0 0 x16 0 0 x190 x22 1 1 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49

Each column gives a point that lies on an (r − 1)-dimensionalsimplexEach set represents one standard-basis vector

Each column gives a point that lies on an (r − 1)-dimensionalsimplex

Each set represents one standard-basis vector

Each column gives a point that lies on an (r − 1)-dimensionalsimplexEach set represents one standard-basis vector

Geometric representation of transversal matroidsEx A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

1 x12 0 0 0 x16 0 0 x190 x22 1 1 x25 x26 x27 0 x290 0 0 0 x35 x36 0 x38 x390 0 0 0 0 0 x47 x48 x49

��

Independent sets in the matroid correspond to affinely independentsets of pointsAll affine relationships are dictated by the faces of the simplexPoints lie on the faces “as freely as possible”

A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)

Where are cyclic flats in the simplex representation?

��

All rank-k cyclic flats lie on (k − 1)-dimensional faces of thesimplex

A flat of a matroid is cyclic if it is a union of circuits (i.e., it has nocoloops)Where are cyclic flats in the simplex representation?

��

Thm (Brylawski 75)A matroid is transversal if and only if it can be represented on asimplex in such a way that every rank-k flat is the set of points ona (k − 1)-dimensional face

(Observe that this bounds the number of cyclic flats transversalmatroids can have)

It is straightforward to go from the set system presentation to theaffine representation on the simplex:

- label vertices of the simplex by sets in the presentation

- place each element x in the span of the vertices correspondingto the sets that contain x

So, if we are given the simplex representation, to recover the setswe just need to look at the complements of the facets

Contractions and duality

I Transversal matroids are not closed under contractions

��

I Transversal matroids are not closed under duality

I Duals of transversal matroids are called cotransversal or strictgammoids. The smallest minor closed class containingtransversal matroids is the class of gammoids

��

Presentations

Transversal matroids typically have several presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 6 7 9},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 7},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 7 9},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 9},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 8 9},C = {5 6 8 9},D = {7 8 9}

Presentations

��

A = {1 2 6 9},B = {2 3 4 5 6 8},C = {5 6 8 9},D = {7 8 9}

Presentations

��

I Aci is a flat

I (Bondy and Welsh 71) a is a coloop of Ac1 if and only if

(A1 ∪ a,A2, . . . ,Ar ) is also a presentation

I (Mason 69, Bondy 72) a transversal matroid has a uniquemaximal presentation (i.e., one in which sets cannot beincreased)

I But there can be several minimal presentations

Presentations

��

I Aci is a flat

Presentations

��

I Aci is a flat

Presentations

��

I Aci is a flat

Presentations

��

I Aci is a flat

Presentations

��

I Aci is a flat

Presentations

��

I Aci is a flat

I Bicircular matroids are those transversal matroids for whichthe simplex representation has points only on vertices andedges

I Deletions of bicircular matroids are bicircular

I Contractions of bicircular matroids are bicircular, if loopless

I Bicircular matroids are not closed under dualityFor instance, U2,n is bicircular but Un−2,n is not if n ≥ 7

I Lattice path matroids are closed under duality

I Lattice path matroids are closed under deletion andcontraction

I The excluded minors for the minor-closed class of lattice pathmatroids are known (Bonin 10)

Characterizations of transversal matroids

Keywords: fundamental transvesal matroids, Mason-Ingletoninequalities, the β function

The Mason-Ingleton characterization

How can one tell if a given matroid is transversal?

Thm (Mason 71, Ingleton 77)The following are equivalent

- M is transversal

- for all non-empty collections F of cyclic flats

( ⋂F∈F

)≤∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

How can one tell if a given matroid is transversal?

- M is transversal

- for all non-empty collections F of cyclic flats

( ⋂F∈F

)≤∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

A transversal matroid M is called fundamental transversal if thereis a basis B = {b1, . . . , br} of M such that in some simplexrepresentation of M the elements b1, . . . , br are placed on vertices

Ex U3,6 is a fundamental transversal matroid

But this one is not

��

But this one is not

��

But this one is not

��

But this one is not

��

Recall that a rank-k cyclic flat always lies on a (k − 1)-dimensionalface of the simplex

Thus, if F is a cyclic flat of a fundamental transversal matroid,then

r(F ) = |B ∩ F |

And similarly, for a collection F of cyclic flats,

( ⋃F∈F

∣∣∣∣∣B ∩ (⋃F∈F

∣∣∣∣∣ =

∣∣∣∣∣ ⋃F∈F

B ∩ F

∣∣∣∣∣r

( ⋂F∈F

∣∣∣∣∣B ∩ (⋂F∈F

∣∣∣∣∣ =

∣∣∣∣∣ ⋂F∈F

B ∩ F

∣∣∣∣∣

Recall that a rank-k cyclic flat always lies on a (k − 1)-dimensionalface of the simplex

Thus, if F is a cyclic flat of a fundamental transversal matroid,then

r(F ) = |B ∩ F |

And similarly, for a collection F of cyclic flats,

( ⋃F∈F

∣∣∣∣∣B ∩ (⋃F∈F

∣∣∣∣∣ =

∣∣∣∣∣ ⋃F∈F

B ∩ F

∣∣∣∣∣r

( ⋂F∈F

∣∣∣∣∣B ∩ (⋂F∈F

∣∣∣∣∣ =

∣∣∣∣∣ ⋂F∈F

B ∩ F

∣∣∣∣∣

Thus, for a fundamental transversal matroid, the Principle ofInclusion and Exclusion gives

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

And if M is not fundamental transversal? We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices

For a cyclic flat F of M, the flat F1 = cl1(F ) is cyclic in M1 andr(F ) = r1(F1)

And for a collection F of cyclic flats of M,

F∈F F)

= r ′(⋃

F1∈F1F1

)r(⋂

F∈F F)≤ r ′

(⋂F1∈F1

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

And if M is not fundamental transversal?

We can extend M to afundamental transversal matroid M1 by adding new elementsa1, . . . , ar on vertices

F∈F F)

= r ′(⋃

F1∈F1F1

)r(⋂

F∈F F)≤ r ′

(⋂F1∈F1

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

F∈F F)

= r ′(⋃

F1∈F1F1

)r(⋂

F∈F F)≤ r ′

(⋂F1∈F1

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

F∈F F)

= r ′(⋃

F1∈F1F1

)r(⋂

F∈F F)≤ r ′

(⋂F1∈F1

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

F∈F F)

= r ′(⋃

F1∈F1F1

)r(⋂

F∈F F)≤ r ′

(⋂F1∈F1

The Mason-Ingleton characterizationWe have proved the ⇓ implication of

- M is transversal

- for all non-empty collection F of cyclic flats

r( ⋂F∈F

F)≤∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

And also the ⇓ implication of

Thm (Bonin, Kung and de Mier 11)The following are equivalent

- M is fundamental transversal

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

The Mason-Ingleton characterizationWe have proved the ⇓ implication of

- M is transversal

r( ⋂F∈F

F)≤∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

And also the ⇓ implication of

Thm (Bonin, Kung and de Mier 11)The following are equivalent

- M is fundamental transversal

r( ⋂F∈F

=∑F ′⊆F

(−1)|F′|+1r

( ⋃F ′∈F ′

F ′)

The β function

What about the implication ⇑ in Mason-Ingleton’s theorem?

Every transvesal matroid has a unique maximal presentation; inthis presentation, the flats Ac

i are actually cyclic flats

Let Z(M) be the set of all cyclic flats of M

Suppose we had a map β : Z(M)→ {0, 1, 2, . . .} such that β(F ) isthe number of times F c appears in the maximal presentation of M

Thus, ∑F∈Z

β(F ) = r(M)

Moreover, as each F must intersect r(F ) sets of the presentation,∑Y∈Z,F∩Y c 6=∅

β(Y ) = r(F )

The β function

Thus, ∑F∈Z

β(F ) = r(M)

β(Y ) = r(F )

The β function

Thus, ∑F∈Z

β(F ) = r(M)

β(Y ) = r(F )

The β function

Thus, ∑F∈Z

β(F ) = r(M)

β(Y ) = r(F )

The β function

Thus, ∑F∈Z

β(F ) = r(M)

β(Y ) = r(F )

The β function

Substracting the previous two equations, the map β should satisfy∑Y∈Z,F⊆Y

β(Y ) = r(M)− r(F )

For any matroid M, define the map β : 2S → Z on all subsets ofelements as

β(X ) = r(M)− r(X )−∑

Y∈Z,X⊂Yβ(Y )

(Ingleton and Piff 72, Mason 71)

Lem If M satisfies the Mason-Ingleton inequalities, thenβ(X ) ≥ 0 for all X

The β function

β(Y ) = r(M)− r(F )

β(X ) = r(M)− r(X )−∑

Y∈Z,X⊂Yβ(Y )

The β function

β(Y ) = r(M)− r(F )

β(X ) = r(M)− r(X )−∑

Y∈Z,X⊂Yβ(Y )

The β function

β(S) = 0, β(X ) = r(M)− r(X )−∑

Y∈Z(M) :X⊂Y β(Y )

��

β({a, b, c}) = 3− 2− 0 = 1

β({a, d , e}) = β({a, f , g}) = 1

β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0

β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1

The β function

β(S) = 0, β(X ) = r(M)− r(X )−∑

��

β({a, b, c}) = 3− 2− 0 = 1

β({a, d , e}) = β({a, f , g}) = 1

β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0

β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1

The β function

β(S) = 0, β(X ) = r(M)− r(X )−∑

��

β({a, b, c}) = 3− 2− 0 = 1

β({a, d , e}) = β({a, f , g}) = 1

β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0

β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1

The β function

β(S) = 0, β(X ) = r(M)− r(X )−∑

��

β({a, b, c}) = 3− 2− 0 = 1

β({a, d , e}) = β({a, f , g}) = 1

β(∅) = 3− 0− (1 + 1 + 1 + 0) = 0

β({a}) = 3− 1− (1 + 1 + 1 + 0) = −1

The β function

It is not very difficult to check that if M is a matroid withβ(X ) ≥ 0 for all X ⊆ S , then the set system

(F c : F ∈ Z, β(F ) > 0)

is a presentation of the matroid M

So all together:

Thm The following are equivalent

- M is transversal

r( ⋂F∈F

F)≤∑F ′⊆F

(−1)|F′|+1r

( ⋂F ′∈F ′

F ′)

- β(X ) ≥ 0 for all subsets X of the ground set

The β function

It is not very difficult to check that if M is a matroid withβ(X ) ≥ 0 for all X ⊆ S , then the set system

(F c : F ∈ Z, β(F ) > 0)

is a presentation of the matroid M

So all together:

Thm The following are equivalent

- M is transversal

r( ⋂F∈F

F)≤∑F ′⊆F

(−1)|F′|+1r

( ⋂F ′∈F ′

F ′)

- β(X ) ≥ 0 for all subsets X of the ground set

Other topics

Keywords: Tutte polynomials of lattice path matroids, transversalextensions, cyclic ordering conjecture

Tutte polynomials of lattice path matroids

Thm (Colbourn, Provan and Vertigan 95)Computing t(M; x , y) is #P-complete for transversal matroids

Can we do better for lattice path matroids?

Thm (Bonin, de Mier and Noy 03)The Tutte polynomial of a lattice path matroid can be computedin polynomial time

Thm (Colbourn, Provan and Vertigan 95)Computing t(M; x , y) is #P-complete for transversal matroids

Can we do better for lattice path matroids?

Recall:

t(M; x , y) =∑

B∈B(M)

x i(B)y e(B)

where i(B), e(B) are the numbers of internally/externally activeelements with respect to basis B and some linear order < on theground set

x 6∈ B is externally active ifwhenever B − b ∪ x is a basis we have x < b

b ∈ B is internally active ifwhenever B − b ∪ x is a basis we have b < x

If B is a basis of a lattice path matroid, can one easily tell whichelements are internally/externally active?

Recall:

t(M; x , y) =∑

B∈B(M)

x i(B)y e(B)

where i(B), e(B) are the numbers of internally/externally activeelements with respect to basis B and some linear order < on theground set

x 6∈ B is externally active ifwhenever B − b ∪ x is a basis we have x < b

b ∈ B is internally active ifwhenever B − b ∪ x is a basis we have b < x

If B is a basis of a lattice path matroid, can one easily tell whichelements are internally/externally active?

Let M be a LPM of rank r and ground set [m + r ]; take the linearorder 1 < 2 < · · ·Let B = {b1 < · · · < bn} be a basis of a lattice path matroid andx 6∈ B; then bi < x < bi+1 for some i

Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally activeWe can use x in place of bi , provided we are not in the lower path!

Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally active

We can use x in place of bi , provided we are not in the lower path!

Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally activeWe can use x in place of bi

, provided we are not in the lower path!

Can we modify the path to include x by removing one step fromb1, . . . , bi? If we cannot, x is externally activeWe can use x in place of bi , provided we are not in the lower path!

Lem An element x 6∈ B is externally active if the E step of Bcorresponding to x belongs to the lower bounding pathDually, an element b ∈ B is internally active if the N step of Bcorresponding to b belongs to the upper bounding path

In other words,

t(M[P,Q]; x , y) =∑π∈P

xQN(π)yPE (π)

where PE (π)/QN(π) are the number of E /N steps of π incommon with P/Q

Lem An element x 6∈ B is externally active if the E step of Bcorresponding to x belongs to the lower bounding pathDually, an element b ∈ B is internally active if the N step of Bcorresponding to b belongs to the upper bounding path

In other words,

t(M[P,Q]; x , y) =∑π∈P

xQN(π)yPE (π)

where PE (π)/QN(π) are the number of E /N steps of π incommon with P/Q

A byproduct:

As turning the paths 180 degrees around is a matroid isomorphismthat switches the paths P and Q, and we should get the sameTutte polynomial no matter how we compute it, one gets that thepairs

(QN(π),PE (π)) and (PN(π),QE (π))

have the same distribution over the paths π ∈ P

(For more on similar results, see Elizalde and Rubey 13+)

A byproduct:

As turning the paths 180 degrees around is a matroid isomorphismthat switches the paths P and Q, and we should get the sameTutte polynomial no matter how we compute it, one gets that thepairs

(QN(π),PE (π)) and (PN(π),QE (π))

have the same distribution over the paths π ∈ P

(For more on similar results, see Elizalde and Rubey 13+)

- Label point (0, 0) with 1

- Recursively, label each point in the region determined by thebounding paths according to

- The label of the point (m, r) is t(M[P,Q]; x , y)

- Label point (0, 0) with 1

- Recursively, label each point in the region determined by thebounding paths according to

f yf f

- The label of the point (m, r) is t(M[P,Q]; x , y)

Transversal extensions

A (single-element) extension of a matroid M on E is a matroid Non E ∪ x such that M = N\x (and, for us, r(N) = r(M))

I The theory of extensions is well-understood: extensions of Mare in bijection with some families of subsets called “modularcuts of flats” (Crapo 65)

A transversal extension of a transversal matroid M is an extensionof M that is also transversal

I If (B1, . . . ,Br ) is a presentation of N, then (B1\x , . . . ,Br\x)is a presentation of M = N\x

I So all transversal extensions of M can be obtained by addingx to some sets in some presentation of M.

I But could it be that we get repetitions? How can we ensurewe have all extensions?

Let A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let

{Ai ∪ x , if i ∈ I ,Ai , otherwise.

The matroid M[AI ] is a transversal extension of M

Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})

Transversal extensionsLet A = (A1, . . . ,Ar ) be a presentation of MFor I ⊆ [r ], let

Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})

I = {1, 2, 3}

Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})

I = {1, 2, 3}

Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})

I = {2, 3}

Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})

I = {3}

Example: A = ({1, 2, 5, 6, 7, 8}, {3, 4, 5, 6, 7}, {7, 8})

I = {1, 2}

Some results from Bonin and de Mier 15

Thm The following are equivalent:

(i) if I 6= J then M[AI ] 6= M[AJ ]

(ii) the presentation A is minimal

Cor If A is a minimal presentation of M, then AI is a minimalpresentation of M[AI ]

Thm If N is a transversal extension of M, there exist a minimalpresentation A of M and a set I ⊆ [r ] such that N = M[AI ]

Let M1,M2 be two matroids on E . The weak order:

M1 ≤w M2 if every independent set in M1 is independent in M2

Fact: the set of all extensions of a matroid M is a lattice under theweak order

Question: is the set of all transversal extensions of a transversalmatroid also a lattice under the weak order?

��

xM M M5 6

Join M5 ∨M6 and meet M5 ∧M6:

��

M5 and M6 are transversal extensions of MBut the ordinary join M5 ∨M6 is not transversal

��

M M M5 6

Join M5 ∨M6 and meet M5 ∧M6:

��

M M M5 6

Transversal join M5 ∨M6 and meet M5 ∧M6:

��

The cyclic ordering problem

Question (Gabow 76)Let B1,B2 be bases of a matroid MIs there an ordering of the elements

b1, b2, . . . , br , br+1, . . . , b2r

such that

- B1 = {b1, . . . , br},- B2 = {br+1, . . . , b2r}, and

- every r cyclically consecutive elements are a basis of M?

It does hold for

I matroids of rank ≤ 4 (Kotlar and Ziv 13)

I graphic matroids (Cordovil and Moreira 93)

I sparse paving matroids (Bonin 13)

I transvesal matroids

b1, b2, . . . , br , br+1, . . . , b2r

such that

- B1 = {b1, . . . , br},- B2 = {br+1, . . . , b2r}, and

It does hold for

b1, b2, . . . , br , br+1, . . . , b2r

such that

- B1 = {b1, . . . , br},- B2 = {br+1, . . . , b2r}, and

It does hold for

It is true for transvesal matroids:

Let M = M[(A1, . . . ,Ar )] and B1,B2 bases of M

Suppose B1 = {b1, . . . , br} with bi ∈ Ai for all iand B2 = {c1, . . . , cr} with ci ∈ Ai for all i

Then every r cyclically consecutive elements of

b1, . . . , br , c1, . . . , cr

form a transversal of (A1, . . . ,Ar ) and thus a basis

(In general, it is true for the larger class of strongly-base oderablematroidsFor more basis-exchange problems, see Bonin 13)

b1, . . . , br , c1, . . . , cr

References from the textEdmonds and Fulkerson 65: Transversals and matroid partition, J. Res.Nat. Bur. Standards Sect. B 69BPiff and Welsh 70: On the vector representations of matroids, J. LondonMath. Soc 2Bonin, de Mier and Noy 03: Lattice path matroids: enumerative aspectsand Tutte polynomials, J. Combin. Theory Ser. A 104Bonin and de Mier 06: Lattice path matroids: structural properties, Eur.J. Combin. 27Simoes-Pereira 72: On subgraphs as matroid cells, Math. Z. 127Matthews 77: Bicircular matroids, Quart. J. Math. 28Brylawski 75: An affine representation for transversal matroids, Studies inAppl. Math. 54Bondy and Welsh 71: Some results on transversal matroids andconstructions for indentically self-dual matroids, Quart. J. Math. 22Bondy 72: Presentations of transversal matroids, J. London Math. Soc 5Mason 69: Representations of Independence Spaces (PhD Dissertation)Bonin 10: Lattice path matroids: the excluded minors J. Combin. TheorySer. B 100Mason 71: A characterization of transversal independence spaces, in:Theorie des Matroıdes (Lecture Notes in Math., 211)

References from the textIngleton 77: Transversal matroids and related structures, in: HigherCombinatorics (Proc. NATO Adv. Study Inst.)Bonin, Kung and de Mier 11: Characterizations of transversal andfundamental transversal matrids, Elec. J. Combin.Ingleton and Piff 72: Gammoids and Transversal Matroids, J. Combin.Theory Ser. B 15Elizalde and Rubey 13: Bijections for pairs of non-crossing lattice pathsand walks in the plane, preprintCrapo 65: Single-element extensions of matroids, J. Res. Nat. Bur.Standards Sect. B 69Bonin and de Mier 15: Extensions and presentations of transversalmatroids, Eur. J. Combin.Gabow 76: Decomposing symmetric exchanges in matroid bases, Math.Programming 10Kotlar and Ziv 13: On serial symmetric exchanges of matroid bases, J.Graph Theory 73Cordovil and Moreira 93: Bases-cobases grahs and polytopes of matroids,Combinatorica 13Bonin 13: Basis-exchange properties of sparse paving matorids, Adv.Appl. Math. 50

Some general references about transversal matroids

J. Bonin, An introduction to transversal matroids (lecture notes online),2010R. Brualdi, Transversal matroids, in Combinatorial geometries, CambridgeUniv. Press, 1987J. Oxley, Matroid Theory, 2n ed, Oxford Univ. Press, 2011 [Sections 1.6,2.4, 11.2]

anna de mier - upc universitat politècnica de catalunya · transversal matroids anna de mier...

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