Feasible Combinatorial Matrix Theory

Ariel G. Fernandez, Michael Soltys.fernanag@mcmaster.ca, soltys@mcmaster.ca

Department of Computing and SoftwareMcMaster University

Hamilton, Ontario, Canada

Outline

IntroductionKMM connects max matching with min vertex core

Language to Formalize Min-Max ReasoningMain Results

LA with ΣB1 -Ind. proves KMM

LA ` Equivalence: Konig, Menger, Hall, Dilworth

Related TheoremsMenger’s Theorem, Hall’s Theorem, and Dilworh’s Theorem

Future Work

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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 aMinimum Vertex Cover.

So by Konig’s Mini-Max Theorem, |M| = |C |.

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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 aMinimum Vertex Cover.

So by Konig’s Mini-Max Theorem, |M| = |C |.

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Language to Formalize Min-Max Reasoning

LA is

I (Developed by Cook and Soltys.) Part of Cook’s program ofReverse Mathematics.

I Three sorts:

I indicesI ring elementsI matrices

I LA formalize linear algebra (Matrix Algebra).I LA over Z (though all matrices are 0-1 matrices.)

I Since we want to count the number of 1s in A by ΣA.

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LA with ΣB1 -Induction

I LA (i.e., LA with ΣB0 -Induction), proves all the ring properties of

matrices (eg.,(AB)C = A(BC )), and LA over Z translates intoTC0-Frege ([Cook-Soltys’04]).

I Bounded Matrix Quantifiers: We let

(∃A ≤ n)α stands for (∃A)[|A| ≤ n ∧ α], and

(∀A ≤ n)α stands for (∀A)[|A| ≤ n→ α].

I LA with ΣB1 -Induction correspond to polytime reasoning and proves

standard properties of the determinant, and translate into extendedFrege.

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Main Results

I Theorem 1:

LA with ΣB1 -Induction ` KMM.

I Theorem 2:

LA proves the equivalence of fundamental theorems:

I Konig Mini-MaxI Menger’s ConnectivityI Hall’s System of Distinct RepresentativesI Dilworth’s Decomposition

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LA with ΣB1 -Ind. proves KMM

Diagonal Property

∗

∗

0

...

0000 . . .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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LA ` Equivalence: Konig, Menger, Hall, Dilworth

Theorem :

LA proves the equivalence of fundamental theorems:

I Konig Mini-Max

I Menger’s Connectivity

I Hall’s System of Distinct Representatives

I Dilworth’s Decomposition

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Menger’s Connectivity Theorem – Example

y

a d

ex b

c f

x y

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Menger’s Connectivity Theorem – Example

y

a d

ex b

c f

x y

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Menger’s Connectivity Theorem – Example

y

a d

ex b

c f

x y

8/12

Menger’s Connectivity Theorem – Example

y

a d

ex b

c f

x y

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Menger’s Connectivity Theorem – Example

y

a d

ex b

c f

x y

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Hall’s SDR Theorem - Example

I Let X = {1, 2, 3, 4, 5} be the 5-set of integers.

I 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).

I Now, if we replace S4 by S ′4 = {2, 5}, then the subsets no

longer have an SDR.

I For S1 ∪ S2 ∪ S ′4 is a 2-set, and three elements are required to

represent S1,S2, S′4

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Dilworth’s Decomposition Theorem - Example

{}

{1} {2} {3}

{1, 2} {1, 3} {2, 3}

{1, 2, 3}

Let P = (⊂, 2X ), i.e., all subsets ofX 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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Examples of LA formalization

For example, concepts necessary to state KMM in LLA:

I Cover(A, α) :=

∀i , j ≤ r(A)(A(i , j) = 1→ α(1, i) = 1 ∨ α(2, j) = 1)

I 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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Future Work

I Can LA-Theory prove KMM?

I What is the relationship between KMM and PHP?(Eg. LA ∪ PHP ` KMM?)

I Can LA ∪KMM prove Hard Matrix Identities?We would like to know whether LA ∪KMM can prove hardmatrix identities, such as AB = I → BA = I . Of course, wealready know from [TZ11] that (non-uniform) NC2-Frege issufficient to prove AB = I → BA = I , and from [Sol06] weknow that ∃LA can prove them also.

I What about ∞-KMM?

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