Beyond Total Unimodularity
Stefan van Zwam
Based on joint work with Rhiannon Hall,
Dillon Mayhew, Rudi Pendavingh,
and Geoff Whittle
EIDMA Seminar, Eindhoven University of Technology, May 4,
2011
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An optimization problem
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Beyond Total Unimodularity
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An optimization problem
minimize 211 + 312 + 413 + 221 + 222 + 823


 


11

4
1 1 1 0 0 0 

 12   
 6
0 0 0 1 1 1 

  

 
such that 1 0 0 1 0 0 ·  13  = 2

 21   
0
1
0
0
1
0

   5
3
0 0 1 0 0 1  22 
23
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Beyond Total Unimodularity
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An optimization problem
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Beyond Total Unimodularity
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An optimization problem
Theorem.
If matrix totally unimodular, then
integer optimal solution.
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Beyond Total Unimodularity
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Totally unimodular matrices
Definition.
A matrix is totally unimodular if
for every square submatrix D
det(D) ∈ {−1, 0, 1}
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Beyond Total Unimodularity
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Testing for total unimodularity
Theorem (Truemper 1982).
There is a polynomial-time algorithm to test
if a matrix A over R is totally unimodular.
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Beyond Total Unimodularity
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Testing for total unimodularity
Theorem (Truemper 1982).
There is a polynomial-time algorithm to test
if a matrix A over R is totally unimodular.
Main ingredient:
Seymour’s Decomposition Theorem
for Regular Matroids(1980).
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Beyond Total Unimodularity
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Overview
Total unimodularity
and its (im)possible generalizations:
I. Matroids and representations
II. Seymour’s Decomposition Theorem
III. Excluded minors
IV. Counting bases
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Beyond Total Unimodularity
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Part I
Matroids and representations
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Beyond Total Unimodularity
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Matroids
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Beyond Total Unimodularity
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Matroids
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Beyond Total Unimodularity
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Matroids
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Beyond Total Unimodularity
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Matroid axioms
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.
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Beyond Total Unimodularity
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Representations
Definition.
A representation of M over field F is a dependencypreserving map
A : E(M) → Fr .
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Beyond Total Unimodularity
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Example: the Fano matroid
 
0
 
0
1
 
1
 
0
1
 
1
 
0
0
 
1
 
1
1
 
0
 
1
1
 
1
 
1
0
 
0
 
1
0
• E = { points }
• I = { X ⊆ E : no 3 on a line, no 4 in a plane}
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Beyond Total Unimodularity
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Representations
Definition.
A representation of M over field F is a dependencypreserving map
A : E(M) → Fr .
View A as r × E matrix!
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Beyond Total Unimodularity
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Regular matroids
Theorem (Tutte 1958).
Equivalent for a matroid M:
• M representable over all fields
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Beyond Total Unimodularity
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Regular matroids
Theorem (Tutte 1958).
Equivalent for a matroid M:
• M representable over all fields
• M representable over GF(2) and GF(3)
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Beyond Total Unimodularity
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Regular matroids
Theorem (Tutte 1958).
Equivalent for a matroid M:
• M representable over all fields
• M representable over GF(2) and GF(3)
• M has totally unimodular representation over R
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Beyond Total Unimodularity
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Regular matroids
Theorem (Tutte 1958).
Equivalent for a matroid M:
• M representable over all fields
• M representable over GF(2) and GF(3)
• M has totally unimodular representation over R
Such matroids are called regular.
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields with characteristic
6= 2
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields with characteristic
6= 2
• M representable over GF(3) and GF(5)
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields with characteristic
6= 2
• M representable over GF(3) and GF(5)
• M has dyadic representation over R
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields with characteristic
6= 2
• M representable over GF(3) and GF(5)
• M has dyadic representation over R
A matrix is dyadic if every subdeterminant is in
{±2k : k ∈ Z} ∪ {0}.
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields except, perhaps,
GF(2)
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields except, perhaps,
GF(2)
• M representable over GF(3), GF(4), GF(5)
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields except, perhaps,
GF(2)
• M representable over GF(3), GF(4), GF(5)
• M representable over GF(3), GF(8)
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields except, perhaps,
GF(2)
• M representable over GF(3), GF(4), GF(5)
• M representable over GF(3), GF(8)
• M has near-regular representation over Q(α)
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Whittle 1997).
Equivalent for a matroid M:
• M representable over all fields except, perhaps,
GF(2)
• M representable over GF(3), GF(4), GF(5)
• M representable over GF(3), GF(8)
• M has near-regular representation over Q(α)
A matrix is near-regular if every subdeterminant is
in
{±α k (1 − α) : k,  ∈ Z} ∪ {0}.
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Beyond Total Unimodularity
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. . . and beyond
Theorem (Vertigan, unpublished; Pendavingh,
vZ 2010).
Equivalent for a matroid M:
• M representable over GF(4), GF(5)
• M has golden ratio representation over R
A matrix is golden ratio if every subdeterminant is in
{±τ k : k ∈ Z} ∪ {0}
where τ is the golden ratio, i.e. τ 2 − τ − 1 = 0.
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Beyond Total Unimodularity
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Part II
Seymour’s Decomposition Theorem
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Beyond Total Unimodularity
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Operations
Elementary operations that preserve T.U.:
• Scale rows and columns by −1
• Permute rows and columns
• Row-reduce a column to an identity vector



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α
c
b
D




→ 
α −1 c
1
0
D − bα −1 c
Beyond Total Unimodularity



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Operations
Dualizing:
[ A] → [−AT  0 ]
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Beyond Total Unimodularity
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Operations
Operations that preserve T.U.: 1-sums
A1 0
A1 ⊕1 A2 =
0 A2
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Beyond Total Unimodularity
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Operations
Operations that preserve T.U.: 2-sums







A1
1
Stefan van Zwam

0
1

.. 

0
 ⊕2  .
0
.
1
0
2
A2



 
 
=
 




Beyond Total Unimodularity

A1
0
1
2
0
A2










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Operations
Operations that preserve T.U.: 3-sums



A 1



 1
b1
0
..
0
1
1
Stefan van Zwam


0 0
1


.. 
1


0 0 ⊕ 3 0

.
1 0
.
0
0 1
1 0
0 1
0 0
..
0 0
 A
1

2
 
b2  
1
 
=
b1
 

A2  

 0

Beyond Total Unimodularity

0
2
b2
A2











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3-sums geometrically
⊕3
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=
Beyond Total Unimodularity
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Have cement, need bricks
Definition.
A matroid is graphic if its independent sets are the
edge sets of forests of a graph G.
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Beyond Total Unimodularity
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Have cement, need bricks
Definition.
A matroid is graphic if its independent sets are the
edge sets of forests of a graph G.
Representation:
{0, 1, −1} with
incidence matrix.
• At most two nonzeroes in each column
• If two, then one 1 and one −1
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Beyond Total Unimodularity
Entries in
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Have cement, need bricks
Definition.
A matroid is graphic if its independent sets are the
edge sets of forests of a graph G.
Representation:
{0, 1, −1} with
incidence matrix.
Entries in
• At most two nonzeroes in each column
• If two, then one 1 and one −1

1
−1

 1

 0
0
2
3
4
5
−1
0
1
0
−1
0
0
1
0
−1
1
0
0
0
−1
1
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6

0

1 

0 
−1
1
4
6
3
2
Beyond Total Unimodularity
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29/53
Have cement, need bricks

1
−1

 1

 0
0
2
3
4
5
−1
0
1
0
−1
0
0
1
0
−1
1
0
0
0
−1
1
6

0

1 

0 
−1
1
4
6
Theorem.
A graphic matroid is regular.
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Beyond Total Unimodularity
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30/53
What else is out there?
Theorem (Seymour 1980).
Suppose M, regular, can not be obtained from
graphic matroids by dualizing, 1-sums, 2-sums.
Then M has one of the following as minor:
R10 , R12
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Beyond Total Unimodularity
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The case R10

−1

 1

 0

 0
1
1
−1
1
0
0
0
1
−1
1
0
0
0
1
−1
1

1

0 

0 
1 

−1
Theorem. If M regular, contains R10 , not equal to
R10 , it can be written as a 1- or 2-sum.
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Beyond Total Unimodularity
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The case R12

1

0

1

0
1

0
0
1
0
1
0
1
1
1
1
0
1
0
1
1
0
1
0
1
0
0
1
−1
1
0

0

0 

1 

−1
0 

−1
Theorem. If M regular and contains R12 , it can be
written as a 3-sum.
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Beyond Total Unimodularity
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Seymour’s Decomposition Theorem
Theorem (Seymour 1980).
Every regular matroid can be obtained from graphic
ones and R10 by dualizing, k-sums for k = 1, 2, 3.
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Beyond Total Unimodularity
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Seymour’s Decomposition Theorem
Theorem (Seymour 1980).
Every regular matroid can be obtained from graphic
ones and R10 by dualizing, k-sums for k = 1, 2, 3.
Theorem (Tutte 1960 + Seymour 1981).
A matroid can be tested for being graphic in polynomial time.
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Beyond Total Unimodularity
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Seymour’s Decomposition Theorem
Theorem (Seymour 1980).
Every regular matroid can be obtained from graphic
ones and R10 by dualizing, k-sums for k = 1, 2, 3.
Theorem (Tutte 1960 + Seymour 1981).
A matroid can be tested for being graphic in polynomial time.
Theorem (Truemper 1982).
A matroid can be tested for being regular in polynomial time.
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Beyond Total Unimodularity
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. . . and beyond?
Problem.
Can a matroid be tested for being near-regular in
polynomial time?
Problem.
Is there a satisfying decomposition theorem for nearregular matroids?
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Beyond Total Unimodularity
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Recognizing signed-graphic matroids
Definition.
A matroid is signed-graphic ⇔
representation over GF(3) with at most 2 nonzero
entries per column.
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Beyond Total Unimodularity
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Recognizing signed-graphic matroids
Definition.
A matroid is signed-graphic ⇔
representation over GF(3) with at most 2 nonzero
entries per column.
Theorem (Geelen; Mayhew – unpublished).
There is no polynomial-time algorithm to test if a
matroid, given by rank oracle, is signed-graphic.
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Beyond Total Unimodularity
35/53
Recognizing signed-graphic matroids
Definition.
A matroid is signed-graphic ⇔
representation over GF(3) with at most 2 nonzero
entries per column.
Theorem (Geelen; Mayhew – unpublished).
There is no polynomial-time algorithm to test if a
matroid, given by rank oracle, is signed-graphic.
But...
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Beyond Total Unimodularity
35/53
Recognizing signed-graphic matroids
Definition.
A matroid is signed-graphic ⇔
representation over GF(3) with at most 2 nonzero
entries per column.
Theorem (Geelen; Mayhew – unpublished).
There is no polynomial-time algorithm to test if a
matroid, given by rank oracle, is signed-graphic.
But...
What if M is given as GF(3)-matrix?
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Beyond Total Unimodularity
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Recognizing signed-graphic matroids
‘Theorem’ (Musitelli 2008).
Polynomial-time algorithm to decide if
A = D−1 A0
with A0 dyadic signed-graphic and D invertible submatrix of A0
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Beyond Total Unimodularity
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Recognizing near-regular-graphic matroids
‘Theorem’ (Pendavingh, vZ unpublished).
Polynomial-time algorithm to decide if ternary matroid is near-regular-graphic.
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Beyond Total Unimodularity
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What about decomposition?
Natural condition for decomposition:
• No basic class contains all graphic and all cographic matroids.
Corollary (Mayhew, Whittle, vZ 2011).
Any natural decomposition of the near-regular matroids must employ 4-sums.
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Beyond Total Unimodularity
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What about decomposition?
Theorem (Mayhew, Whittle, vZ 2011).
M1 , M2 graphic matroids. Can build internally 4connected near-regular matroid having both M1 and
dual of M2 in it.
d e ƒ 4 5 6 

1 0 1 1 1 0


b  0 −1 1 1 0 α 


c  1 1 0 0 α −α 
A12 = 

1 0 0 0 1 0 1 


2  0 0 0 0 1 −1 
3
0 0 0 1 1 0
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Beyond Total Unimodularity
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Part III
Excluded minors
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Beyond Total Unimodularity
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Minors: abstract definition
• Deletion: M\e := (E − {e}, { ∈ I : e 6∈ })
• Contraction: M/ e := (E − {e}, { :  ∪ {e} ∈ I })
• Minors: Obtained from sequence of such steps
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Beyond Total Unimodularity
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Minors: abstract definition
• Deletion: M\e := (E − {e}, { ∈ I : e 6∈ })
• Contraction: M/ e := (E − {e}, { :  ∪ {e} ∈ I })
• Minors: Obtained from sequence of such steps
É Generate partial order
É Preserve representability
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Beyond Total Unimodularity
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Excluded minors
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Beyond Total Unimodularity
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Regular matroids
Theorem (Tutte 1958):
Exactly 3 excluded minors for regular matroids,
namely
¨
,
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,
Beyond Total Unimodularity
∗
«
44/53
Near-regular matroids
Theorem (Hall, Mayhew, vZ 2011):
Exactly 10 excluded minors for near-regular matroids,
Stefan van Zwam
Beyond Total Unimodularity
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namely
¨
,
,
,
,
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∗
,
∗
,
∗ ,
Beyond Total Unimodularity
,
ΔY
«
,
46/53
...and beyond?
Problem.
Is there a finite number of excluded minors for
dyadic matroids?
Stefan van Zwam
Beyond Total Unimodularity
46/53
...and beyond?
Problem.
Is there a finite number of excluded minors for
dyadic matroids?
Matroid Minors Project
Whittle, in progress).
Yes! (And an algorithm too!)
Stefan van Zwam
(Geelen,
Beyond Total Unimodularity
Gerards,
47/53
Part IV
Counting bases
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Beyond Total Unimodularity
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Regular matroids
Definition.
Matrix A over R
is totally unimodular if each nonsingular square submatrix D has
| det(D)| = 1.
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Beyond Total Unimodularity
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Regular matroids
Definition.
Matrix A over R
is totally unimodular if each nonsingular square submatrix D has
| det(D)| = 1.
Matrix Tree Theorem (Kirchhoff 1847).
det(AAT ) = #{B : B basis of M[A]}.
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Beyond Total Unimodularity
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Sixth-roots-of-unity matroids
Definition.
Matrix A over C
is complex unimodular if each nonsingular square
submatrix D has
| det(D)| = 1.
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Beyond Total Unimodularity
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Sixth-roots-of-unity matroids
Definition.
Matrix A over C
is complex unimodular if each nonsingular square
submatrix D has
| det(D)| = 1.
Theorem (Choe, Oxley, Sokal, Wagner 2004).
Complex-unimodular matroid is sixth-roots-ofunity:
has representation with all entries in
{0, 1, ζ, ζ2 , . . . , ζ5 }.
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Beyond Total Unimodularity
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Sixth-roots-of-unity matroids
Definition.
Matrix A over C
is complex unimodular if each nonsingular square
submatrix D has
| det(D)| = 1.
Theorem (Choe, Oxley, Sokal, Wagner 2004).
Complex-unimodular matroid is sixth-roots-ofunity:
has representation with all entries in
{0, 1, ζ, ζ2 , . . . , ζ5 }.
Theorem (Choe, Oxley, Sokal, Wagner 2004).
det(AA† ) = #{B : B basis of M[A]}.
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Beyond Total Unimodularity
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Quaternionic unimodular matroids
Definition.
Matrix A over H
is quaternionic unimodular if each nonsingular
square submatrix D has
δ(D) = 1.
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Beyond Total Unimodularity
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Quaternionic unimodular matroids
Definition.
Matrix A over H
is quaternionic unimodular if each nonsingular
square submatrix D has
δ(D) = 1.
Theorem (Pendavingh, vZ, in preparation).
δ(AA† ) = #{B : B basis of M[A]}.
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Beyond Total Unimodularity
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Questions
Can we count the bases of other classes efficiently?
• #P-hard in general
• Dyadic: {0} ∪ {±2z : z ∈ Z}
• 2-uniform: {0} ∪ {±α j (1 − α)k β (1 − β)m (α − β)n :
j, k, , m, n ∈ Z}
• k-uniform
• ...
Stefan van Zwam
Beyond Total Unimodularity
51/53
Questions
Can we count the bases of other classes efficiently?
• #P-hard in general
• Dyadic: {0} ∪ {±2z : z ∈ Z}
• 2-uniform: {0} ∪ {±α j (1 − α)k β (1 − β)m (α − β)n :
j, k, , m, n ∈ Z}
• k-uniform
• ...
Do quaternionic unimodular matroids have the halfplane property?
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Beyond Total Unimodularity
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Where to learn more?
• Matroïden en hun representaties, Nieuw
Archief voor Wiskunde, December 2010
• (With Rudi Pendavingh) Lifts of matroid representations over partial fields, Journal of Combinatorial Theory, Series B, vol. 100, Issue 1, pp.
36-67, 2010
• (With Rhiannon Hall and Dillon Mayhew) The excluded minors for near-regular matroids, European Journal of Combinatorics, in press, 2011
• (With Dillon Mayhew and Geoff Whittle) An obstacle to a decomposition theorem for nearregular matroids, SIAM Journal on Discrete
Mathematics, in press, 2011
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Beyond Total Unimodularity
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Slides, preprints at http://www.cwi.nl/~zwam/
The End
Stefan van Zwam
Beyond Total Unimodularity