laten zien dat het niet past/ how to show it doesn’t ﬁtsvanzwam/talks/20090416.pdf · laten...

Where innovation starts

Laten zien dat het niet past/How to show it doesn’t fit

Stefan van Zwam

Based on joint work with Rhiannon Hall and Dillon Mayhew

Nederlands Mathematisch Congres, April 15, 2009

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/ department of mathematics and computer science

Matroid theory

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Linearly independent vectors in Rn

Example

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Example

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Example

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Lemma. Given

E: finite set of vectors

I: collection of linearly independent subsets

then

•∅ ∈ I• J ∈ I and ⊆ J, then ∈ I• , J ∈ I and || < |J|, then

∃e ∈ J \ such that ∪ {e} ∈ I

Matroid axioms

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Definition. Given

E: finite set

I: collection of subsets

such that

•∅ ∈ I• J ∈ I and ⊆ J, then ∈ I• , J ∈ I and || < |J|, then

∃e ∈ J \ such that ∪ {e} ∈ I

Then M = (E,I) is a matroid.

Matroid axioms

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1899: Hilbert, Bernays: Plane geometry

1900–1936: Dedekind, Birkhoff, MacLane:Semimodular lattices

1910–1937: Steinitz, Van der Waerden:Algebraic dependence

1936: Nakasawa: Projective geometry

1935: Whitney: Lin. dependence, duality, graphs

1942: Rado: Transversals (matching theory)

1958: Tutte: Connectivity, minors, . . .

1971: Edmonds: Greedy algorithm

Rota, Brylawski, Seymour, . . .

Matroids everywhere!

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The representation problem

Problem. Is there a map

E→ Fn

preserving the dependencies of M = (E,I)?

Matroid representation

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Example: the Fano matroid

100

001

010

• E = { points }

• I = {X ⊆ E in general position }


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100

001

010

• E = { points }



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100

001

010

• E = { points }



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100

001

010

• E = { points }



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100

001

010

101

011

111

110

• E = { points }



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100

001

010

101

011

111

110

110

−101

−011

= −002


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How to show it doesn’t fit?

Problem. Is there a dependency-preserving map

E(M)→GF(2)

F

...

?


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E(M)→GF(2)

F

...

?

• “Yes” certified by vectors {1, . . . , n}


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E(M)→GF(2)

F

...

?

• “Yes” certified by vectors {1, . . . , n}

• How to certify “no”?


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Reducing a set of vectors: deletion

Matroid minors

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Reducing a set of vectors: deletion

Matroid minors

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Reducing a set of vectors: projection

Matroid minors

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Matroid minors

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Matroid minors

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Abstract definition• Deletion: M\e := (E \ {e},{ ∈ I : e 6∈ })• Contraction: M/e := (E \ {e},{ : ∪ {e} ∈ I})• Minors: Obtained from sequence of such steps

Matroid minors

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Abstract definition• Deletion: M\e := (E \ {e},{ ∈ I : e 6∈ })• Contraction: M/e := (E \ {e},{ : ∪ {e} ∈ I})• Minors: Obtained from sequence of such steps

– Generate partial order– Preserve representability

Matroid minors

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Excluded minors

Matroid minors

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That’s how!

Problem:Is there a dependency-preserving map

¨M : E(M)→

GF(2)

F

...

«

• How to certify the answer is “no”?

• By reducing to an excluded minor!


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That’s how!

Problem:Is there a dependency-preserving map

¨M : E(M)→

GF(2)

F

...

«

• How to certify the answer is “no”?

• By reducing to an excluded minor!

• Rota’s Conjecture: finitely many


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Rota’s Conjecture

Conjecture (Rota 1971): F finite, then ∃k = k(F) :exactly k excluded minors for

¨M : E(M)→

GF(2)

F

...

«


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Rota’s Conjecture

Conjecture (Rota 1971): F finite, then ∃k = k(F) :exactly k excluded minors for

¨M : E(M)→

GF(2)

F

...

«

• Proven for F ∈ {GF(2),GF(3),GF(4)}


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Regular matroids

Theorem (Tutte 1958):Exactly 3 excluded minors for

¨M : E(M)→

GF(2)GF(3)GF(4)GF(5)GF(7)...

«


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Near-regular matroids

Theorem (Hall, Mayhew, vZ 2009):Exactly 10 excluded minors for

¨M : E(M)→

GF(2)GF(3)GF(4)GF(5)GF(7)...

«


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Excluded minors for near-regular¨, , ,

� �∗,

,� �∗

, ,

,� �∗

,� �ΔY«


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¨M : E(M)→

GF(2)GF(3)GF(4)GF(5)GF(7)...

«


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¨M : E(M)→

GF(2)

P

...

«


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¨M : E(M)→

GF(2)

P

...

«Why care?• Complexity of algorithms

• Structure of ternary matroids

• Sharpening the knives for Rota’s Conjecture


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Summary

• Matroid axioms abstract linear dependence

• Matroids occur everywhere

• Representation problem

• Excluded minors

• Near-regular matroids

Thank you for listening

Preprint at http://www.win.tue.nl/~svzwam/

The end

http://www.win.tue.nl/~svzwam/

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