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

On the K¨ onig–Hall theorem for multidimensional matrices Anna Taranenko Sobolev Institute of Mathematics, Novosibirsk, Russia Groups and Graphs, Designs and Dynamics Yichang, China, 2019 Anna Taranenko Multidimensional K¨ onig–Hall theorem 1 / 23

K¨ onig–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 . Anna Taranenko Multidimensional K¨ onig–Hall theorem 2 / 23

K¨ onig–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¨ onig–Hall theorem 2 / 23

K¨ onig–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 0 0 0     0 0 0 0 1 0 0 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 2 / 23

K¨ onig–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 0 0 0     0 0 0 0 1 0 0 0 positive partial diagonal ↔ polyplex lines covering unity entries ↔ hyperplane cover Anna Taranenko Multidimensional K¨ onig–Hall theorem 2 / 23

Multidimensional matrices Let n , d ∈ N , and let I d n = { ( α 1 , . . . , α d ) : α i ∈ { 1 , . . . , n }} . A d -dimensional matrix A of order n is an array ( a α ) α ∈ I d n , a α ∈ R . Anna Taranenko Multidimensional K¨ onig–Hall theorem 3 / 23

Multidimensional matrices Let n , d ∈ N , and let I d n = { ( α 1 , . . . , α d ) : α i ∈ { 1 , . . . , n }} . A d -dimensional matrix A of order n is an array ( a α ) α ∈ I d n , a α ∈ R . A matrix A is nonnegative if a α ≥ 0 for all α ∈ I d n . Anna Taranenko Multidimensional K¨ onig–Hall theorem 3 / 23

Multidimensional matrices Let n , d ∈ N , and let I d n = { ( α 1 , . . . , α d ) : α i ∈ { 1 , . . . , n }} . A d -dimensional matrix A of order n is an array ( a α ) α ∈ I d n , a α ∈ R . A matrix A is nonnegative if a α ≥ 0 for all α ∈ I d n . A hyperplane Γ in a d -dimensional matrix is a maximal ( d − 1)-dimensional submatrix. Anna Taranenko Multidimensional K¨ onig–Hall theorem 3 / 23

Multidimensional matrices Let n , d ∈ N , and let I d n = { ( α 1 , . . . , α d ) : α i ∈ { 1 , . . . , n }} . A d -dimensional matrix A of order n is an array ( a α ) α ∈ I d n , a α ∈ R . A matrix A is nonnegative if a α ≥ 0 for all α ∈ I d n . A hyperplane Γ in a d -dimensional matrix is a maximal ( d − 1)-dimensional submatrix.  0 0 0 0 0 1 0 1 0  0 0 1 0 0 0 1 0 0   0 1 0 1 0 0 0 0 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 3 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α Anna Taranenko Multidimensional K¨ onig–Hall theorem 4 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α 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). Anna Taranenko Multidimensional K¨ onig–Hall theorem 4 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α 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. Anna Taranenko Multidimensional K¨ onig–Hall theorem 4 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α 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 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1   1 0 0 0 0 0 1 1 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 4 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α 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 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1   1 0 0 0 0 0 1 1 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 5 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α 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 0 0 1 1 0 1 1 0 0 1 1 / 2 1 / 2 0 0 1   1 0 0 0 0 0 1 / 2 1 / 2 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 6 / 23

Polyplexes and diagonals A polyplex of weight W is a nonnegative multidimensional matrix K such that � k α ≤ 1 for each hyperplane Γ and � k α = W . α ∈ Γ α 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 0 0 1 1 0 1 1 1 0 0 1 / 2 1 / 2 0 1 1   1 0 0 1 0 0 1 1 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 7 / 23

Hyperplane covers A hyperplane cover of weight W of a d -dimensional (0 , 1)-matrix A of order 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 � λ i , j ≥ 1 and � λ i , j = W . Γ i , j ∋ α i , j Anna Taranenko Multidimensional K¨ onig–Hall theorem 8 / 23

Hyperplane covers A hyperplane cover of weight W of a d -dimensional (0 , 1)-matrix A of order 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 � λ i , j ≥ 1 and � λ i , j = W . Γ i , j ∋ α i , j A hyperplane cover Λ of a matrix A is optimal if it has a minimum weight among all hyperplane covers of A . Anna Taranenko Multidimensional K¨ onig–Hall theorem 8 / 23

Hyperplane covers A hyperplane cover of weight W of a d -dimensional (0 , 1)-matrix A of order 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 � λ i , j ≥ 1 and � λ i , j = W . Γ i , j ∋ α i , j 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. Anna Taranenko Multidimensional K¨ onig–Hall theorem 8 / 23

Hyperplane covers A hyperplane cover of weight W of a d -dimensional (0 , 1)-matrix A of order 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 � λ i , j ≥ 1 and � λ i , j = W . Γ i , j ∋ α i , j 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.     1 1 1 1 1 1 1 0 0 1 1 / 2 0 A = 1 1 1 1 0 0 0 0 0 Λ = 1 0 0     1 1 1 1 0 0 0 0 0 1 / 2 0 0 Anna Taranenko Multidimensional K¨ onig–Hall theorem 8 / 23