On the K onigHall theorem for multidimensional matrices Anna - - PowerPoint PPT Presentation

On the K¨

• nig–Hall theorem for multidimensional

matrices

Anna Taranenko

Sobolev Institute of Mathematics, Novosibirsk, Russia

Groups and Graphs, Designs and Dynamics Yichang, China, 2019

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• nig–Hall theorem

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K¨

• nig–Hall theorem

The maximal length of a positive partial diagonal in a (0, 1)-matrix A is equal to the minimal number of lines covering all unity entries of A.

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• nig–Hall theorem

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K¨

• nig–Hall theorem

The maximal length of a positive partial diagonal in a (0, 1)-matrix A is equal to the minimal number of lines covering all unity entries of A. n-ordered matrices extremal for containing a unity diagonal are exactly the matrices whose zero entries form a (r × t)-rectangle with r + t = n + 1.

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

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K¨

• nig–Hall theorem

The maximal length of a positive partial diagonal in a (0, 1)-matrix A is equal to the minimal number of lines covering all unity entries of A. n-ordered matrices extremal for containing a unity diagonal are exactly the matrices whose zero entries form a (r × t)-rectangle with r + t = n + 1. Extremal matrices of order 4     1 1 1 1 1 1 1 1 1 1 1 1         1 1 1 1 1 1 1 1 1 1    

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• nig–Hall theorem

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K¨

• nig–Hall theorem

The maximal length of a positive partial diagonal in a (0, 1)-matrix A is equal to the minimal number of lines covering all unity entries of A. n-ordered matrices extremal for containing a unity diagonal are exactly the matrices whose zero entries form a (r × t)-rectangle with r + t = n + 1. Extremal matrices of order 4     1 1 1 1 1 1 1 1 1 1 1 1         1 1 1 1 1 1 1 1 1 1     positive partial diagonal ↔ polyplex lines covering unity entries ↔ hyperplane cover

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Multidimensional matrices

Let n, d ∈ N, and let I d

n = {(α1, . . . , αd) : αi ∈ {1, . . . , n}}.

A d-dimensional matrix A of order n is an array (aα)α∈I d

n , aα ∈ R. Anna Taranenko Multidimensional K¨

• nig–Hall theorem

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Multidimensional matrices

Let n, d ∈ N, and let I d

n = {(α1, . . . , αd) : αi ∈ {1, . . . , n}}.

A d-dimensional matrix A of order n is an array (aα)α∈I d

n , aα ∈ R.

A matrix A is nonnegative if aα ≥ 0 for all α ∈ I d

n .

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• nig–Hall theorem

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Multidimensional matrices

Let n, d ∈ N, and let I d

n = {(α1, . . . , αd) : αi ∈ {1, . . . , n}}.

A d-dimensional matrix A of order n is an array (aα)α∈I d

n , aα ∈ R.

A matrix A is nonnegative if aα ≥ 0 for all α ∈ I d

n .

A hyperplane Γ in a d-dimensional matrix is a maximal (d − 1)-dimensional submatrix.

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Multidimensional matrices

Let n, d ∈ N, and let I d

n = {(α1, . . . , αd) : αi ∈ {1, . . . , n}}.

A d-dimensional matrix A of order n is an array (aα)α∈I d

n , aα ∈ R.

A matrix A is nonnegative if aα ≥ 0 for all α ∈ I d

n .

A hyperplane Γ in a d-dimensional matrix is a maximal (d − 1)-dimensional submatrix.   1 1 1 1 1 1  

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• nig–Hall theorem

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W .

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• nig–Hall theorem

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W . A polyplex of weight n and order n is a polydiagonal. A diagonal is a (0, 1)-matrix with exactly one unity entry in each hyperplane (i.e. the simplest polydiagonal).

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• nig–Hall theorem

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W . A polyplex of weight n and order n is a polydiagonal. A diagonal is a (0, 1)-matrix with exactly one unity entry in each hyperplane (i.e. the simplest polydiagonal). An optimal polyplex in a (0, 1)-matrix A is a polyplex of a maximum weight in the matrix.

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W . A polyplex of weight n and order n is a polydiagonal. A diagonal is a (0, 1)-matrix with exactly one unity entry in each hyperplane (i.e. the simplest polydiagonal). An optimal polyplex in a (0, 1)-matrix A is a polyplex of a maximum weight in the matrix. Matrix A   1 1 1 1 1 1 1 1 1 1 1 1 1 1  

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• nig–Hall theorem

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W . A polyplex of weight n and order n is a polydiagonal. A diagonal is a (0, 1)-matrix with exactly one unity entry in each hyperplane (i.e. the simplest polydiagonal). An optimal polyplex in a (0, 1)-matrix A is a polyplex of a maximum weight contained in the matrix. Diagonal in the matrix A   1 1 1 1 1 1 1 1 1 1 1 1 1 1  

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• nig–Hall theorem

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W . A polyplex of weight n and order n is a polydiagonal. A diagonal is a (0, 1)-matrix with exactly one unity entry in each hyperplane (i.e. the simplest polydiagonal). An optimal polyplex in a (0, 1)-matrix A is a polyplex of a maximum weight contained in the matrix. Polydiagonal in the matrix A  

1/2

1

1/2

1 1 1 1

1/2 1/2

1 1 1

1/2 1/2

 

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Polyplexes and diagonals

A polyplex of weight W is a nonnegative multidimensional matrix K such that

α∈Γ

kα ≤ 1 for each hyperplane Γ and

α

kα = W . A polyplex of weight n and order n is a polydiagonal. A diagonal is a (0, 1)-matrix with exactly one unity entry in each hyperplane (i.e. the simplest polydiagonal). An optimal polyplex in a (0, 1)-matrix A is a polyplex of a maximum weight contained in the matrix. Polyplex of weight 2 in matrix A   1 1 1 1 1 1 1

1/2 1/2

1 1 1 1 1 1 1  

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Hyperplane covers

A hyperplane cover of weight W of a d-dimensional (0, 1)-matrix A of

• rder n is a (d × n)-table Λ assigning nonnegative weights to all

hyperplanes of A so that for each α from the support of A it holds

• Γi,j∋α

λi,j ≥ 1 and

i,j

λi,j = W .

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• nig–Hall theorem

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Hyperplane covers

A hyperplane cover of weight W of a d-dimensional (0, 1)-matrix A of

• rder n is a (d × n)-table Λ assigning nonnegative weights to all

hyperplanes of A so that for each α from the support of A it holds

• Γi,j∋α

λi,j ≥ 1 and

i,j

λi,j = W . A hyperplane cover Λ of a matrix A is optimal if it has a minimum weight among all hyperplane covers of A.

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• nig–Hall theorem

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Hyperplane covers

A hyperplane cover of weight W of a d-dimensional (0, 1)-matrix A of

• rder n is a (d × n)-table Λ assigning nonnegative weights to all

hyperplanes of A so that for each α from the support of A it holds

• Γi,j∋α

λi,j ≥ 1 and

i,j

λi,j = W . A hyperplane cover Λ of a matrix A is optimal if it has a minimum weight among all hyperplane covers of A. Given a nonnegative table Λ, we define the (0, 1)-matrix A(Λ) such that the support of A consists of all entries covered with weight at least 1 by Λ.

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• nig–Hall theorem

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Hyperplane covers

A hyperplane cover of weight W of a d-dimensional (0, 1)-matrix A of

• rder n is a (d × n)-table Λ assigning nonnegative weights to all

hyperplanes of A so that for each α from the support of A it holds

• Γi,j∋α

λi,j ≥ 1 and

i,j

λi,j = W . A hyperplane cover Λ of a matrix A is optimal if it has a minimum weight among all hyperplane covers of A. Given a nonnegative table Λ, we define the (0, 1)-matrix A(Λ) such that the support of A consists of all entries covered with weight at least 1 by Λ. A =   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1

1/2

1

1/2

 

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Hyperplane covers

A hyperplane cover of weight W of a d-dimensional (0, 1)-matrix A of

• rder n is a (d × n)-table Λ assigning nonnegative weights to all

hyperplanes of A so that for each α from the support of A it holds

• Γi,j∋α

λi,j ≥ 1 and

i,j

λi,j = W . A hyperplane cover Λ of a matrix A is optimal if it has a minimum weight among all hyperplane covers of A. Given a nonnegative table Λ, we define the (0, 1)-matrix A(Λ) such that the support of A consists of all entries covered with weight at least 1 by Λ. A =   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   = A(Λ); Λ =   1

1/2 1/2 1/2

 

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Hypergraph interpretation

A d-dimensional (0, 1)-matrix of order n is the adjacency matrix of parts

• f a d-uniform d-partite hypergraph with parts of size n.

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Hypergraph interpretation

A d-dimensional (0, 1)-matrix of order n is the adjacency matrix of parts

• f a d-uniform d-partite hypergraph with parts of size n.

A fractional matching in a hypergraph is an assignment of nonnegative weights to hyperedges such that each vertex is covered with total weight at most 1. A fractional matching in a d-partite hypergraph is a polyplex in the d-dimensional adjacency matrix of parts.

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Hypergraph interpretation

A d-dimensional (0, 1)-matrix of order n is the adjacency matrix of parts

• f a d-uniform d-partite hypergraph with parts of size n.

A fractional matching in a hypergraph is an assignment of nonnegative weights to hyperedges such that each vertex is covered with total weight at most 1. A fractional matching in a d-partite hypergraph is a polyplex in the d-dimensional adjacency matrix of parts. A fractional transversal in a hypergraph H is an assignment of nonnegative weights to vertices such that each hyperedge is covered with total weight at least 1. A fractional transversal in a d-partite hypergraph is a hyperplane cover of the d-dimensional adjacency matrix of parts.

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Linear programming approach

Given a d-dimensional (0, 1)-matrix A of order n, maximal weight of a polyplex in A and minimal weight of a hyperplane cover of A are dual linear programming problems.

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Linear programming approach

Given a d-dimensional (0, 1)-matrix A of order n, maximal weight of a polyplex in A and minimal weight of a hyperplane cover of A are dual linear programming problems.

Corollary

The maximal weight of a polyplex in a matrix is equal to the minimal weight of its hyperplane cover.

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Linear programming approach

Given a d-dimensional (0, 1)-matrix A of order n, maximal weight of a polyplex in A and minimal weight of a hyperplane cover of A are dual linear programming problems.

Corollary

The maximal weight of a polyplex in a matrix is equal to the minimal weight of its hyperplane cover.

Theorem

Let A be a d-dimensional (0, 1)-matrix of order n, Λ be its optimal hyperplane cover, and K be an optimal polyplex in A.

1 All nonzero entries of K are covered with weight 1 by Λ. 2 If a hyperplane Γ is taken with a nonzero weight in Λ then the sum of

entries of K in the hyperplane Γ equals 1.

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Extremal matrices

A d-dimensional (0, 1)-matrix A is extremal if it has no polydiagonals and after replacing any 0-entry by 1-entry a polydiagonal appears.

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Extremal matrices

A d-dimensional (0, 1)-matrix A is extremal if it has no polydiagonals and after replacing any 0-entry by 1-entry a polydiagonal appears. The deficiency δ of an extremal matrix A of order n is the difference between n and the weight of the optimal polyplex in A.

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Extremal matrices

A d-dimensional (0, 1)-matrix A is extremal if it has no polydiagonals and after replacing any 0-entry by 1-entry a polydiagonal appears. The deficiency δ of an extremal matrix A of order n is the difference between n and the weight of the optimal polyplex in A.

K¨

• nig–Hall theorem for 2-dimensional matrices

All 2-dimensional extremal matrices have deficiency δ = 1 and are defined by their optimal hyperplane covers.

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• nig–Hall theorem

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Multidimensional K¨

• nig–Hall theorem

Complete characterization of all extremal multidimensional matrices in terms of their deficiencies and optimal hyperplane covers.

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• nig–Hall theorem

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Multidimensional K¨

• nig–Hall theorem

Complete characterization of all extremal multidimensional matrices in terms of their deficiencies and optimal hyperplane covers. 3-dimensional extremal matrices of order 2 1. 1 1 1 1

• Λ =

  1   δ = 1.

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• nig–Hall theorem

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Multidimensional K¨

• nig–Hall theorem

Complete characterization of all extremal multidimensional matrices in terms of their deficiencies and optimal hyperplane covers. 3-dimensional extremal matrices of order 2 1. 1 1 1 1

• Λ =

  1   δ = 1. 2. 1 1 1 1

• Λ =

 

1/2 1/2 1/2

  δ = 1/2.

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3-dimensional extremal matrices of order 3

1.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1 1   δ = 1. 2.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1 1   δ = 1.

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3-dimensional extremal matrices of order 3

1.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1 1   δ = 1. 2.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1 1   δ = 1. 3.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1

1/2 1/2 1/2

  δ = 1/2. 4.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =  

1/2 1/2 1/2 1/2 1/2

  δ = 1/2.

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3-dimensional extremal matrices of order 3

1.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1 1   δ = 1. 2.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1 1   δ = 1. 3.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =   1

1/2 1/2 1/2

  δ = 1/2. 4.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =  

1/2 1/2 1/2 1/2 1/2

  δ = 1/2. 5.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =  

2/3 1/3 2/3 1/3 1/3 1/3

  δ = 1/3. 6.   1 1 1 1 1 1 1 1 1 1 1 1 1 1 1   Λ =  

3/4 1/2 1/2 1/4 1/2 1/4

  δ = 1/4.

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Extremal matrices and optimal hyperplane covers

Question

Does there exist an equivalence between extremal matrices and their

• ptimal hyperplane covers?

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• nig–Hall theorem

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Extremal matrices and optimal hyperplane covers

Question

Does there exist an equivalence between extremal matrices and their

• ptimal hyperplane covers?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A then A = A(Λ).

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Extremal matrices and optimal hyperplane covers

Question

Does there exist an equivalence between extremal matrices and their

• ptimal hyperplane covers?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A then A = A(Λ).

Conjecture

Every extremal matrix has the unique optimal hyperplane cover.

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Extremal matrices and optimal hyperplane covers

Question

Does there exist an equivalence between extremal matrices and their

• ptimal hyperplane covers?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A then A = A(Λ).

Conjecture

Every extremal matrix has the unique optimal hyperplane cover. Uniqueness of an optimal hyperplane cover for every extremal matrix is equivalent to existence of a unique solution of linear programming problems of certain types.

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Uniqueness of optimal hyperplane cover

Proposition

If an optimal hyperplane cover Λ of a matrix A covers all upper 1-entries with weight 1 then Λ is the unique optimal hyperplane cover.

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Uniqueness of optimal hyperplane cover

Proposition

If an optimal hyperplane cover Λ of a matrix A covers all upper 1-entries with weight 1 then Λ is the unique optimal hyperplane cover.

Theorem

If A is one of the following extremal matrices then it has the unique

• ptimal hyperplane cover.

1 A has deficiency 1, 1/2 or 1/3. 2 A has an optimal hyperplane cover Λ with all entries λi,j ∈ {0, λ} for

some λ.

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Deficiencies of extremal matrices

Question

What are the possible values of deficiencies δ of extremal matrices?

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Deficiencies of extremal matrices

Question

What are the possible values of deficiencies δ of extremal matrices?

Proposition

For the deficiency δ of an extremal matrices it holds 0 < δ ≤ 1.

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Deficiencies of extremal matrices

Question

What are the possible values of deficiencies δ of extremal matrices?

Proposition

For the deficiency δ of an extremal matrices it holds 0 < δ ≤ 1.

Conjecture

The deficiency δ of any extremal matrix is equal to 1/m for some m ∈ N. If Λ is an optimal hyperplane cover of an extremal matrix then all weights λi,j are integer multiples of the deficiency δ.

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Conditions on optimal hyperplane covers

Question

What conditions on a table Λ guarantee that it defines a multidimensional matrix with an optimal polyplex of the same weight?

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Conditions on optimal hyperplane covers

Question

What conditions on a table Λ guarantee that it defines a multidimensional matrix with an optimal polyplex of the same weight?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A of deficiency δ then the following hold:

1 each row of Λ contains at least one zero entry; Anna Taranenko Multidimensional K¨

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Conditions on optimal hyperplane covers

Question

What conditions on a table Λ guarantee that it defines a multidimensional matrix with an optimal polyplex of the same weight?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A of deficiency δ then the following hold:

1 each row of Λ contains at least one zero entry; 2 there are no indices covered by Λ with weight strictly between 1 − δ

and 1;

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Conditions on optimal hyperplane covers

Question

What conditions on a table Λ guarantee that it defines a multidimensional matrix with an optimal polyplex of the same weight?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A of deficiency δ then the following hold:

1 each row of Λ contains at least one zero entry; 2 there are no indices covered by Λ with weight strictly between 1 − δ

and 1;

3 the difference between non-equal entries λi,j and λi,l from the same

row of Λ is not less than δ;

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Conditions on optimal hyperplane covers

Question

What conditions on a table Λ guarantee that it defines a multidimensional matrix with an optimal polyplex of the same weight?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A of deficiency δ then the following hold:

1 each row of Λ contains at least one zero entry; 2 there are no indices covered by Λ with weight strictly between 1 − δ

and 1;

3 the difference between non-equal entries λi,j and λi,l from the same

row of Λ is not less than δ;

4 for every weight λi,j = 0 or 1 we have δ ≤ λi,j ≤ 1 − δ; Anna Taranenko Multidimensional K¨

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Conditions on optimal hyperplane covers

Question

What conditions on a table Λ guarantee that it defines a multidimensional matrix with an optimal polyplex of the same weight?

Theorem

If Λ is an optimal hyperplane cover of an extremal matrix A of deficiency δ then the following hold:

1 each row of Λ contains at least one zero entry; 2 there are no indices covered by Λ with weight strictly between 1 − δ

and 1;

3 the difference between non-equal entries λi,j and λi,l from the same

row of Λ is not less than δ;

4 for every weight λi,j = 0 or 1 we have δ ≤ λi,j ≤ 1 − δ; 5 the number of λi,j > 1/2 is less than n. Anna Taranenko Multidimensional K¨

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Extremal matrices of big deficiencies

Theorem

1 There is a one-to-one correspondence between nonequivalent

d-dimensional extremal matrices of order n and of deficiency 1 and the Young diagrams with n − 1 cells and no more than d rows.

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Extremal matrices of big deficiencies

Theorem

1 There is a one-to-one correspondence between nonequivalent

d-dimensional extremal matrices of order n and of deficiency 1 and the Young diagrams with n − 1 cells and no more than d rows.

2 There are no extremal matrices with deficiency 1/2 < δ < 1.

Λ is the optimal hyperplane cover of a d-dimensional extremal matrix

• f order n and deficiency 1/2 if and only if all λi,j ∈ {0, 1/2, 1}, weight
• f Λ is n − 1/2 and the number of 1/2-entries in each row of Λ is less

than the number of 1/2-entries in the union of all other rows.

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Extremal matrices of big deficiencies

Theorem

1 There is a one-to-one correspondence between nonequivalent

d-dimensional extremal matrices of order n and of deficiency 1 and the Young diagrams with n − 1 cells and no more than d rows.

2 There are no extremal matrices with deficiency 1/2 < δ < 1.

Λ is the optimal hyperplane cover of a d-dimensional extremal matrix

• f order n and deficiency 1/2 if and only if all λi,j ∈ {0, 1/2, 1}, weight
• f Λ is n − 1/2 and the number of 1/2-entries in each row of Λ is less

than the number of 1/2-entries in the union of all other rows.

3 There are no extremal matrices with deficiency 1/3 < δ < 1/2.

If δ = 1/3 then for all optimal hyperplane covers Λ of extremal matrices of deficiencies 1/3 we have all λi,j ∈ {0, 1/3, 2/3, 1}.

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Two-value optimal hyperplane covers

Theorem

1 If a hyperplane cover Λ of weight n − 1/m and with entries

λi,j ∈ {0, 1/m} has at least one zero in each row then Λ is the unique

• ptimal hyperplane cover of the d-dimensional extremal matrix

A = A(Λ) of order n and of deficiency δ = 1/m.

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Two-value optimal hyperplane covers

Theorem

1 If a hyperplane cover Λ of weight n − 1/m and with entries

λi,j ∈ {0, 1/m} has at least one zero in each row then Λ is the unique

• ptimal hyperplane cover of the d-dimensional extremal matrix

A = A(Λ) of order n and of deficiency δ = 1/m.

2 There are no other extremal matrices with two-value optimal

hyperplane covers.

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Diagonal extremality of extremal matrices

A multidimensional (0, 1)-matrix A is diagonally extremal if it has no polydiagonals and after replacing any 0-entry by 1-entry a diagonal appears.

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Diagonal extremality of extremal matrices

A multidimensional (0, 1)-matrix A is diagonally extremal if it has no polydiagonals and after replacing any 0-entry by 1-entry a diagonal appears.

Conjecture

Every extremal matrix is diagonally extremal.

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Diagonal extremality of extremal matrices

A multidimensional (0, 1)-matrix A is diagonally extremal if it has no polydiagonals and after replacing any 0-entry by 1-entry a diagonal appears.

Conjecture

Every extremal matrix is diagonally extremal.

Conjecture for hypergraphs

Every d-partite d-uniform hypergraph with parts of equal sizes extremal for fractional perfect matchings is also extremal for integer perfect matchings.

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

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Diagonal extremality of extremal matrices

Theorem

All known constructions of extremal matrices preserve diagonal extremality.

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

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Diagonal extremality of extremal matrices

Theorem

All known constructions of extremal matrices preserve diagonal extremality.

Theorem

If A is an extremal matrix defined by a hyperplane cover Λ whose weights have only two values then the matrix A is diagonally extremal. The proof is based on the Gale–Ryser theorem for (0, 1)-matrices.

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

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Diagonal extremality of extremal matrices

Theorem

All known constructions of extremal matrices preserve diagonal extremality.

Theorem

If A is an extremal matrix defined by a hyperplane cover Λ whose weights have only two values then the matrix A is diagonally extremal. The proof is based on the Gale–Ryser theorem for (0, 1)-matrices.

Corollary

Every extremal matrix of deficiency 1 or 1/2 is diagonally extremal.

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

22 / 23

Diagonal extremality of extremal matrices

Theorem

All known constructions of extremal matrices preserve diagonal extremality.

Theorem

If A is an extremal matrix defined by a hyperplane cover Λ whose weights have only two values then the matrix A is diagonally extremal. The proof is based on the Gale–Ryser theorem for (0, 1)-matrices.

Corollary

Every extremal matrix of deficiency 1 or 1/2 is diagonally extremal.

Theorem

Every multidimensional extremal matrix of order 2 is diagonally extremal.

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

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See preprint https://arxiv.org/abs/1811.09981 for

1 more properties and constructions of extremal matrices and their

• ptimal hyperplane covers;

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

23 / 23

See preprint https://arxiv.org/abs/1811.09981 for

1 more properties and constructions of extremal matrices and their

• ptimal hyperplane covers;

2 list of all extremal matrices and their optimal hyperplane covers of

small orders and dimensions;

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

23 / 23

See preprint https://arxiv.org/abs/1811.09981 for

1 more properties and constructions of extremal matrices and their

• ptimal hyperplane covers;

2 list of all extremal matrices and their optimal hyperplane covers of

small orders and dimensions;

3 more conjectures and relations to different topics. Anna Taranenko Multidimensional K¨

• nig–Hall theorem

23 / 23

See preprint https://arxiv.org/abs/1811.09981 for

1 more properties and constructions of extremal matrices and their

• ptimal hyperplane covers;

2 list of all extremal matrices and their optimal hyperplane covers of

small orders and dimensions;

3 more conjectures and relations to different topics.

Thank you for your attention!

Anna Taranenko Multidimensional K¨

• nig–Hall theorem

23 / 23