Feasible Combinatorial Matrix Theory Ariel G. Fern´ andez, Michael Soltys. [email protected], [email protected] Department of Computing and Software McMaster University Hamilton, Ontario, Canada
Feasible Combinatorial Matrix Theory
Ariel G. Fernandez, Michael [email protected], [email protected]
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
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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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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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