This presentation was given by Ariel Fernandez as a short presentation at LICS'13, June 28, 2013, at Tulane University in New Orleans.

1. Feasible Combinatorial Matrix Theory
Ariel G. Fern´andez, Michael Soltys.
fernanag@mcmaster.ca, soltys@mcmaster.ca
Department of Computing and Software
McMaster University
Hamilton, Ontario, Canada

2. Outline
Introduction
KMM connects max matching with min vertex core
Language to Formalize Min-Max Reasoning
Main Results
LA with ΣB
1 -Ind. proves KMM
LA Equivalence: K¨onig, Menger, Hall, Dilworth
Related Theorems
Menger’s Theorem, Hall’s Theorem, and Dilworh’s Theorem
Future Work
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3. KMM connects max matching with min vertex core
1
2
3
4
5
1’
2’
3’
4’
V1 V2
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4. KMM connects max matching with min vertex core
1
2
3
4
5
1’
2’
3’
4’
V1 V2
M is a Matching denoted by snaked lines.
C is a Vertex cover denoted by square nodes.
Here M is a Maximum Matching and V is a
Minimum Vertex Cover.
So by K¨onig’s Mini-Max Theorem, |M| = |C|.
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5. Language to Formalize Min-Max Reasoning
LA is
(Developed by Cook and Soltys.) Part of Cook’s program of
Reverse Mathematics.
Three sorts:
indices
ring elements
matrices
LA formalize linear algebra (Matrix Algebra).
LA over Z (though all matrices are 0-1 matrices.)
Since we want to count the number of 1s in A by ΣA.
3/12

6. LA with ΣB
1 -Induction
LA (i.e., LA with ΣB
0 -Induction), proves all the ring properties of
matrices (eg.,(AB)C = A(BC)), and LA over Z translates into
TC0
-Frege ([Cook-Soltys’04]).
Bounded Matrix Quantifiers: We let
(∃A ≤ n)α stands for (∃A)[|A| ≤ n ∧ α], and
(∀A ≤ n)α stands for (∀A)[|A| ≤ n → α].
LA with ΣB
1 -Induction correspond to polytime reasoning and proves
standard properties of the determinant, and translate into extended
Frege.
4/12

7. Main Results
Theorem 1:
LA with ΣB
1 -Induction KMM.
Theorem 2:
LA proves the equivalence of fundamental theorems:
K¨onig Mini-Max
Menger’s Connectivity
Hall’s System of Distinct Representatives
Dilworth’s Decomposition
5/12

8. LA with ΣB
1 -Ind. proves KMM
Diagonal Property
∗
∗
0
...
00
0
0 . . .1
Either Aii = 1 or (∀j ≥ i)[Aij = 0 ∧ Aji = 0].
Claim Given any matrix A, ∃LA proves that there exist permutation
matrices P, Q such that PAQ has the diagonal property.
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9. LA Equivalence: K¨onig, Menger, Hall, Dilworth
Theorem :
LA proves the equivalence of fundamental theorems:
K¨onig Mini-Max
Menger’s Connectivity
Hall’s System of Distinct Representatives
Dilworth’s Decomposition
7/12

10. Menger’s Connectivity Theorem – Example
y
a d
ex b
c f
x y

11. Menger’s Connectivity Theorem – Example
y
a d
ex b
c f
x y
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12. Menger’s Connectivity Theorem – Example
y
a d
ex b
c f
x y
8/12

13. Menger’s Connectivity Theorem – Example
y
a d
ex b
c f
x y
8/12

14. Menger’s Connectivity Theorem – Example
y
a d
ex b
c f
x y
8/12

15. Hall’s SDR Theorem - Example
Let X = {1, 2, 3, 4, 5} be the 5-set of integers.
Let S = {S1, S2, S3, S4} be a family of X. For instance,
S1 = {2, 5}, S2 = {2, 5}, S3 = {1, 2, 3, 4}, S4 = {1, 2, 5}.
Then D := (2, 5, 3, 1) is an SDR for (S1, S2, S3, S4).
Now, if we replace S4 by S4 = {2, 5}, then the subsets no
longer have an SDR.
For S1 ∪ S2 ∪ S4 is a 2-set, and three elements are required to
represent S1, S2, S4
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16. Dilworth’s Decomposition Theorem - Example
{}
{1} {2} {3}
{1, 2} {1, 3} {2, 3}
{1, 2, 3}
Let P = (⊂, 2X
), i.e., all subsets of
X with |X| = n with set inclusion,
x < y ⇐⇒ x ⊂ y.
(A) Suppose that the largest
chain in P has size . Then P can
be partitioned into antichains.
We have 4-antichains [{}] ,
[{1}, {2}, {3}] , [{1, 2}, {1, 3}, {2, 3}] ,
and [{1, 2, 3}] .
(B) Suppose that the largest
antichain in P has size .
Then P can be partitioned into
disjoint chains. We have
[{} ⊂ {1} ⊂ {1, 2} ⊂ {1, 2, 3}] ,
[{2} ⊂ {2, 3}] , and [{3} ⊂ {1, 3}].
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17. Examples of LA formalization
For example, concepts necessary to state KMM in LLA:
Cover(A, α) :=
∀i, j ≤ r(A)(A(i, j) = 1 → α(1, i) = 1 ∨ α(2, j) = 1)
Select(A, β) :=
∀i, j ≤ r(A)((β(i, j) = 1 → A(i, j) = 1)
∧
∀k ≤ r(A)(β(i, j) = 1 → β(i, k) = 0 ∧ β(k, j) = 0))
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18. Future Work
Can LA-Theory prove KMM?
What is the relationship between KMM and PHP?
(Eg. LA ∪ PHP KMM?)
Can LA ∪ KMM prove Hard Matrix Identities?
We would like to know whether LA ∪ KMM can prove hard
matrix identities, such as AB = I → BA = I. Of course, we
already know from [TZ11] that (non-uniform) NC2
-Frege is
sufficient to prove AB = I → BA = I, and from [Sol06] we
know that ∃LA can prove them also.
What about ∞-KMM?
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