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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
2/31
/ department of mathematics and computer science
Matroid theory
3/31
/ department of mathematics and computer science
Linearly independent vectors in Rn
Example
4/31
/ department of mathematics and computer science
Linearly independent vectors in Rn
Example
5/31
/ department of mathematics and computer science
Linearly independent vectors in Rn
Example
6/31
/ department of mathematics and computer science
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
7/31
/ department of mathematics and computer science
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
8/31
/ department of mathematics and computer science
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!
9/31
/ department of mathematics and computer science
The representation problem
Problem. Is there a map
E→ Fn
preserving the dependencies of M = (E,I)?
Matroid representation
10/31
/ department of mathematics and computer science
Example: the Fano matroid
100
001
010
• E = { points }
• I = {X ⊆ E in general position }
Matroid representation
11/31
/ department of mathematics and computer science
Example: the Fano matroid
100
001
010
• E = { points }
• I = {X ⊆ E in general position }
Matroid representation
12/31
/ department of mathematics and computer science
Example: the Fano matroid
100
001
010
• E = { points }
• I = {X ⊆ E in general position }
Matroid representation
13/31
/ department of mathematics and computer science
Example: the Fano matroid
100
001
010
• E = { points }
• I = {X ⊆ E in general position }
Matroid representation
14/31
/ department of mathematics and computer science
Example: the Fano matroid
100
001
010
101
011
111
110
• E = { points }
• I = {X ⊆ E in general position }
Matroid representation
15/31
/ department of mathematics and computer science
Example: the Fano matroid
100
001
010
101
011
111
110
110
−101
−011
= −002
Matroid representation
16/31
/ department of mathematics and computer science
How to show it doesn’t fit?
Problem. Is there a dependency-preserving map
E(M)→GF(2)
F
...
?
Matroid representation
16/31
/ department of mathematics and computer science
How to show it doesn’t fit?
Problem. Is there a dependency-preserving map
E(M)→GF(2)
F
...
?
• “Yes” certified by vectors {1, . . . , n}
Matroid representation
16/31
/ department of mathematics and computer science
How to show it doesn’t fit?
Problem. Is there a dependency-preserving map
E(M)→GF(2)
F
...
?
• “Yes” certified by vectors {1, . . . , n}
• How to certify “no”?
Matroid representation
17/31
/ department of mathematics and computer science
Reducing a set of vectors: deletion
Matroid minors
18/31
/ department of mathematics and computer science
Reducing a set of vectors: deletion
Matroid minors
19/31
/ department of mathematics and computer science
Reducing a set of vectors: projection
Matroid minors
20/31
/ department of mathematics and computer science
Reducing a set of vectors: projection
Matroid minors
21/31
/ department of mathematics and computer science
Reducing a set of vectors: projection
Matroid minors
22/31
/ department of mathematics and computer science
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
22/31
/ department of mathematics and computer science
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
23/31
/ department of mathematics and computer science
Excluded minors
Matroid minors
24/31
/ department of mathematics and computer science
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!
Matroid representation
24/31
/ department of mathematics and computer science
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
Matroid representation
25/31
/ department of mathematics and computer science
Rota’s Conjecture
Conjecture (Rota 1971): F finite, then ∃k = k(F) :exactly k excluded minors for
¨M : E(M)→
GF(2)
F
...
«
Matroid representation
25/31
/ department of mathematics and computer science
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)}
Matroid representation
26/31
/ department of mathematics and computer science
Regular matroids
Theorem (Tutte 1958):Exactly 3 excluded minors for
¨M : E(M)→
GF(2)GF(3)GF(4)GF(5)GF(7)...
«
Matroid representation
27/31
/ department of mathematics and computer science
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)...
«
Matroid representation
28/31
/ department of mathematics and computer science
Excluded minors for near-regular¨, , ,
� �∗,
,� �∗
, ,
,� �∗
,� �ΔY«
Matroid representation
29/31
/ department of mathematics and computer science
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)...
«
Matroid representation
30/31
/ department of mathematics and computer science
Near-regular matroids
Theorem (Hall, Mayhew, vZ 2009):Exactly 10 excluded minors for
¨M : E(M)→
GF(2)
P
...
«
Matroid representation
30/31
/ department of mathematics and computer science
Near-regular matroids
Theorem (Hall, Mayhew, vZ 2009):Exactly 10 excluded minors for
¨M : E(M)→
GF(2)
P
...
«Why care?• Complexity of algorithms
• Structure of ternary matroids
• Sharpening the knives for Rota’s Conjecture
Matroid representation
31/31
/ department of mathematics and computer science
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
